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Seismic fragility of RC frame and wall-frame dual buildings designed ...

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<strong>Seismic</strong> <strong>fragility</strong> <strong>of</strong> <strong>RC</strong> <strong>frame</strong> <strong>and</strong> <strong>wall</strong>-<strong>frame</strong><strong>dual</strong> <strong>buildings</strong> <strong>designed</strong> to EN- EurocodesA Dissertation Submitted in Partial Fulfilment <strong>of</strong> the Requirementsfor the Master Degree inEarthquake Engineering &/or Engineering SeismologyByKyriakos AntoniouSupervisor(s): Pr<strong>of</strong>. Michael N. FardisFebruary, 2013University <strong>of</strong> Patras


The dissertation entitled “<strong>Seismic</strong> <strong>fragility</strong> <strong>of</strong> <strong>RC</strong> <strong>frame</strong> <strong>and</strong> <strong>wall</strong>-<strong>frame</strong> <strong>dual</strong> <strong>buildings</strong><strong>designed</strong> to EN-Eurocodes” by Kyriakos Antoniou, has been approved in partial fulfilment <strong>of</strong>the requirements for the Master Degree in Earthquake EngineeringPr<strong>of</strong>essor M. N. Fardis ________________


AbstractABSTRACTFragility curves are constructed for structural members <strong>of</strong> regular reinforced concrete <strong>frame</strong><strong>and</strong> <strong>wall</strong>-<strong>frame</strong> <strong>buildings</strong> <strong>designed</strong> according to Eurocode 2 <strong>and</strong> Eurocode 8. Prototype plan<strong>and</strong>height- wise very regular <strong>buildings</strong> are studied with parameters including the height <strong>of</strong> thebuilding, the level <strong>of</strong> Eurocode 8 design (in terms <strong>of</strong> design peak ground acceleration <strong>and</strong>ductility class) <strong>and</strong> for <strong>dual</strong> systems the percentage <strong>of</strong> seismic base shear taken by the <strong>wall</strong>s.Member <strong>fragility</strong> curves are constructed based on the results <strong>of</strong> nonlinear static (pushover)analysis (SPO) <strong>and</strong> incremental dynamic analysis (IDA) using 14 spectrum-compatible semiartificialaccelerograms. Analysis is performed using three-dimensional structural models <strong>of</strong>the full <strong>buildings</strong>. These results are compared to <strong>fragility</strong> curves obtained from previousstudies for a simplified analysis method using the lateral force method (LFM).The <strong>fragility</strong> curves are addressed on two member limit states; yielding <strong>and</strong> the ultimatedeformation in bending or shear. The peak chord rotation <strong>and</strong> peak shear force dem<strong>and</strong>s atmember ends are taken as damage measures; the peak ground acceleration (PGA) is used asseismic intensity measure. The probability <strong>of</strong> exceedance <strong>of</strong> each limit state is computed fromthe probability distributions <strong>of</strong> the damage measures (conditional on intensity measure) <strong>and</strong> <strong>of</strong>the corresponding capacities.The alternative methods yield results that are in good agreement for beams <strong>and</strong> columns inboth <strong>frame</strong> <strong>and</strong> <strong>dual</strong> <strong>buildings</strong> <strong>and</strong> for the flexural behavior <strong>of</strong> <strong>wall</strong>s. Results from thesimplified procedure using the LFM shows that Medium Ductility Class <strong>wall</strong>s are likely t<strong>of</strong>ail in shear even before their design PGA. The dynamic analysis confirms to a certain extendthe inelastic amplification <strong>of</strong> shear forces due to higher mode effects <strong>and</strong> shows that therelevant rules <strong>of</strong> Eurocode 8 are on the conservative side.Keywords: Concrete <strong>buildings</strong>; Concrete <strong>wall</strong>s; Eurocode 8; Fragility curves; <strong>Seismic</strong> Design1


AcknowledgementsACKNOWLEDGEMENTSI would like to sincerely thank my supervisor Pr<strong>of</strong>essor M. N. Fardis for his guidance <strong>and</strong> thetime dedicated <strong>and</strong> G. Tsionis for his continuous support for the project.2


IndexTABLE OF CONTENTSABSTRACT ............................................................................................................................................ 1ACKNOWLEDGEMENTS ..................................................................................................................... 2TABLE OF CONTENTS......................................................................................................................... 3LIST OF FIGURES ................................................................................................................................. 6LIST OF TABLES ................................................................................................................................. 12LIST OF SYMBOLS ............................................................................................................................. 141. INTRODUCTION ........................................................................................................................ 202. DEFINITIONS AND BACKGROUND ....................................................................................... 222.1. Building codes ..........................................................................................................222.2. Performance-based requirements ..............................................................................222.3. Intensity Measure ......................................................................................................232.4. Damage measures .....................................................................................................252.5. <strong>Seismic</strong> Vulnerability Assessment Methodologies ...................................................262.5.1. Empirical Fragility Curves ................................................................................262.5.2. Expert Opinion method .....................................................................................272.5.3. Analytical Fragility Curves ...............................................................................282.5.4. Hybrid methods .................................................................................................302.6. <strong>Seismic</strong> safety assessment <strong>of</strong> <strong>RC</strong> <strong>buildings</strong> <strong>designed</strong> to EC8...................................303. DESCRIPTION OF BUILDINGS ................................................................................................ 323.1. Typology <strong>of</strong> <strong>buildings</strong> ...............................................................................................323


Index3.2. Geometry <strong>of</strong> <strong>buildings</strong> ..............................................................................................323.3. Materials ...................................................................................................................344. DESIGN OF BUILDINGS ........................................................................................................... 354.1. Actions on structure <strong>and</strong> assumptions.......................................................................354.2. Behaviour factors <strong>and</strong> local ductility ........................................................................364.3. Design procedure ......................................................................................................374.3.1. Sizing <strong>of</strong> beams <strong>and</strong> columns in <strong>frame</strong> systems ...............................................374.3.2. Sizing <strong>of</strong> beams, columns <strong>and</strong> <strong>wall</strong>s in <strong>wall</strong>-<strong>frame</strong> (<strong>dual</strong>) systems ..................384.4. Dimensioning <strong>of</strong> Beams ............................................................................................394.5. Dimensioning <strong>of</strong> Columns ........................................................................................404.6. Dimensioning <strong>of</strong> Walls .............................................................................................425. ANALYSIS METHODS AND MODELLING ASSUMPTIONS ................................................ 465.1. Nonlinear Static “Pushover” Analysis ......................................................................465.2. Incremental Dynamic Analysis .................................................................................475.3. Structural modelling for IDA <strong>and</strong> SPO .....................................................................515.4. Linear Static Analysis - “Lateral Force Method” .....................................................536. ASSESMENT OF BUILDINGS ................................................................................................... 576.1. Limit State <strong>of</strong> Damage Limitation (DL) ...................................................................576.2. Limit State <strong>of</strong> Near Collapse (NC) ...........................................................................606.3. Estimation <strong>of</strong> damage measure dem<strong>and</strong>s ..................................................................637. METHODOLOGY OF FRAGILITY ANALYSIS ....................................................................... 647.1. Damage Measures .....................................................................................................647.2. Exclusion <strong>of</strong> unrealistic results for IDA ...................................................................657.3. Determination <strong>of</strong> variability ......................................................................................657.4. Construction <strong>of</strong> <strong>fragility</strong> curves ................................................................................698. RESULTS AND DISCUSSION ................................................................................................... 7148.1. Modal analysis results ...............................................................................................728.2. Median PGAs at attainment <strong>of</strong> the damage state for the three methods ...................748.3. Fragility curve results for <strong>wall</strong>-<strong>frame</strong> <strong>dual</strong> systems ..................................................768.4. Fragility curve results for <strong>frame</strong> systems ..................................................................918.5. Comparison between analysis methods ....................................................................968.6. Fragility results <strong>of</strong> <strong>wall</strong>s in the ultimate state .........................................................111


Index9. SUMMARY AND CONCLUSIONS ......................................................................................... 116REFERENCES .................................................................................................................................... 119APPENDIX A ....................................................................................................................................... A1APPENDIX B ....................................................................................................................................... B1APPENDIX C ....................................................................................................................................... C15


IndexLIST OF FIGURESFIGURE 2.1 DEFINITION OF CHORD ROTATION [ADAPTED FROM FARDIS, 2009] ................................... 26FIGURE 2.2 FLOWCHART TO DESCRIBE THE COMPONENTS OF THE CALCULATION OF ANALYTICALVULNERABILITY CURVE [ADAPTED FROM DUMOVA-JOVANOSKA (2004)]................................... 29FIGURE 3.1 PLAN OF WALL-FRAME (DUAL) BUILDINGS [PAPAILIA, 2011] ............................................ 33FIGURE 3.2 GEOMETRY OF FRAME BUILDINGS [PAPAILIA, 2011] .......................................................... 33FIGURE 3.3 STRUCTURAL 3D MODEL TAKEN FROM ANSRUOP FOR FIVE – STOREY DUAL SYSTEM ...... 34FIGURE 4.1CAPACITY DESIGN VALUES OF SHEAR FO<strong>RC</strong>ES ON BEAMS [CEN, 2004] .............................. 40FIGURE 4.2 CAPACITY DESIGN SHEAR FO<strong>RC</strong>E IN COLUMNS [CEN 2004] ............................................... 42FIGURE 4.3: DESIGN ENVELOPE FOR BENDING MOMENTS IN THE SLENDER WALLS (LEFT: WALLSYSTEMS ; RIGHT: DUAL SYSTEMS ) [CEN 2004] .......................................................................... 43FIGURE 4.4 DESIGN ENVELOPE OF THE SHEAR FO<strong>RC</strong>ES IN THE WALLS OF A DUAL SYSTEM [CEN 2004]....................................................................................................................................................... 44FIGURE 5.1 PSEUDO-ACCELERATION SPECTRA FOR THE SEMI-ARTIFICIAL INPUT MOTIONS COMPAREDTO THE SMOOTH TARGET SPECTRUM (SHOWN WITH THICK BLACK LINE) ..................................... 49FIGURE 5.2 TIME-HISTORIES OF ACCELEROGRAMS USED IN THE ANALYSIS .......................................... 50FIGURE 5.3 TAKEDA MODEL MODIFIED BY LITTON AND OTANI ............................................................ 51FIGURE 5.4 STRUCTURAL MODEL FOR A FIVE – STOREY DUAL BUILDING TAKEN FROM ANSRUOP ..... 53FIGURE 5.5 STRUCTURAL MODEL FOR AN EIGHT – STOREY DUAL BUILDING TAKEN FROM ANSRUOP 53FIGURE 7.1 EXCLUSION OF UNREALISTIC RESULTS IN IDA (DAMAGE INDICES ABOVE CONTINUOUSLINE ARE NEGLECTED) .................................................................................................................. 65FIGURE 7.2 COEFFICIENT OF VARIATION (COV) OF DM-DEMANDS FOR FIVE-STOREY FRAME BUILDINGDESIGNED TO DC M AND PGA=0.20G .......................................................................................... 676


IndexFIGURE 7.3 COEFFICIENT OF VARIATION (COV) OF DM-DEMANDS FOR FIVE-STOREY FRAME-EQUIVALENT BUILDING DESIGNED TO DC M AND PGA=0.20G .................................................... 68FIGURE 8.1 FRAGILITY CURVES FOR FIVE-STOREY WALL-EQUIVALENT BUILDING DESIGNED TOPGA=0.20G AND DC M ANALYZED USING IDA METHOD ............................................................ 77FIGURE 8.2 FRAGILITY CURVES OF WALLS FOR EIGHT-STOREY FRAME-EQUIVALENT (LEFT) AND WALL-EQUIVALENT BUILDING (RIGHT) DESIGNED TO PGA=0.20G AND DC M ANALYZED USING IDAMETHOD ......................................................................................................................................... 78FIGURE 8.3 FRAGILITY CURVES OF WALLS FOR EIGHT-STOREY FRAME-EQUIVALENT (LEFT) AND WALL-EQUIVALENT BUILDING (RIGHT) DESIGNED TO PGA=0.25G AND DC M ANALYZED USING IDAMETHOD ......................................................................................................................................... 78FIGURE 8.4 FRAGILITY CURVES OF WALLS FOR FIVE-STOREY FRAME-EQUIVALENT BUILDING DESIGNEDTO PGA=0.20G AND DC M (LEFT) AND WALL BUILDING DESIGNED TO DC H AND PGA=0.25G(RIGHT) ANALYZED USING IDA METHOD ...................................................................................... 78FIGURE 8.5 FRAGILITY CURVES OF WALLS FOR FIVE-STOREY FRAME-EQUIVALENT (LEFT) AND WALL-EQUIVALENT (RIGHT) BUILDINGS DESIGNED TO DC H AND PGA=0.25G ANALYZED USING IDAMETHOD ......................................................................................................................................... 79FIGURE 8.6 BEAM FRAGILITY CURVES FOR A) YIELDING AND B) ULTIMATE STATE OF A FIVE-STOREYFRAME-EQUIVALENT (LEFT), WALL-EQUIVALENT (MIDDLE) AND WALL SYSTEM (RIGHT)BUILDING DESIGNED TO DC M AND PGA=0.20G ANALYZED WITH IDA ...................................... 80FIGURE 8.7 COLUMN FRAGILITY CURVES FOR C) YIELDING AND D) ULTIMATE STATE OF A FIVE-STOREYFRAME-EQUIVALENT (LEFT), WALL-EQUIVALENT (MIDDLE) AND WALL SYSTEM (RIGHT)BUILDING DESIGNED TO DC M AND PGA=0.20G ANALYZED WITH IDA ...................................... 80FIGURE 8.8 BEAM FRAGILITY CURVES FOR A) YIELDING AND B) ULTIMATE STATE OF A EIGHT-STOREYFRAME-EQUIVALENT (LEFT), WALL-EQUIVALENT (MIDDLE) AND WALL SYSTEM (RIGHT)BUILDING DESIGNED TO DC M AND PGA=0.25G ANALYZED WITH IDA ...................................... 81FIGURE 8.9 COLUMN FRAGILITY CURVES FOR C) YIELDING AND D) ULTIMATE STATE OF A EIGHT-STOREY FRAME-EQUIVALENT (LEFT), WALL-EQUIVALENT (MIDDLE) AND WALL SYSTEM (RIGHT)BUILDING DESIGNED TO DC M AND PGA=0.25G ANALYZED WITH IDA ...................................... 81FIGURE 8.10 FRAGILITY CURVES FOR MOST CRITICAL MEMBERS OF FIVE–STOREY FRAME-EQUIVALENTBUILDING DESIGNED TO PGA=0.25G AND DC M ANALYZED USING IDA METHOD...................... 83FIGURE 8.11 FRAGILITY CURVES FOR MOST CRITICAL MEMBERS OF FIVE–STOREY WALL-EQUIVALENTBUILDING DESIGNED TO PGA=0.25G AND DC M ANALYZED USING IDA METHOD...................... 84FIGURE 8.12 FRAGILITY CURVES FOR MOST CRITICAL MEMBERS OF FIVE–STOREY WALL BUILDINGDESIGNED TO PGA=0.25G AND DC M ANALYZED USING IDA METHOD ...................................... 857


IndexFIGURE 8.13 MEMBER FRAGILITY CURVES OF FRAME-EQUIVALENT DUAL SYSTEMS DESIGNED TOPGA=0.25G AND DC M FOR: (TOP) FIVE – STOREY; (BOTTOM) EIGHT-STOREY USING IDAMETHOD ......................................................................................................................................... 86FIGURE 8.14 MEMBER FRAGILITY CURVES FOR WALL SYSTEMS DESIGNED TO PGA=0.25G AND DC MCURVES OF: (TOP) FIVE – STOREY; (BOTTOM) EIGHT-STOREY USING IDA METHOD ...................... 87FIGURE 8.15 MEMBER FRAGILITY CURVES FOR A FIVE-STOREY FRAME-EQUIVALENT (FE), WALL-EQUIVALENT (WE), WALL DUAL (WS) SYSTEM DESIGNED TO PGA=0.20G AND DC M USING SPOMETHOD FOR MOST CRITICAL STOREY MEMBERS. ........................................................................ 88FIGURE 8.16 FRAGILITY CURVES OF EIGHT–STOREY FRAME-EQUIVALENT BUILDING DESIGNED TO DCM AND: (TOP) PGA=0.20G; (BOTTOM) PGA=0.25G ANALYZED USING IDA METHOD ................. 89FIGURE 8.17 MEMBER FRAGILITY CURVES FOR A EIGHT-STOREY WALL-EQUIVALENT SYSTEMDESIGNED TO DC M AND FOR PGA=0.20G AND PGA=0.25G USING IDA METHOD FOR MOSTCRITICAL STOREY MEMBERS. ........................................................................................................ 90FIGURE 8.18 MEMBER FRAGILITY CURVES FOR A FIVE-STOREY FRAME SYSTEM DESIGNED DC M ANDTO PGA=0.20G AND PGA=0.25G USING IDA METHOD FOR MOST CRITICAL STOREY MEMBERS. 91FIGURE 8.19 MEMBER FRAGILITY CURVES FOR A FIVE-STOREY FRAME SYSTEM DESIGNED PGA=0.25GAND TO DC M AND DC H USING IDA METHOD FOR MOST CRITICAL STOREY MEMBERS. ............ 92FIGURE 8.20 FRAGILITY CURVES OF FIVE-STOREY BUILDINGS DESIGNED TO PGA=0.25G AND DC MANALYZED USING IDA METHOD: (TOP) FRAME BUILDINGS; (BOTTOM) FRAME-EQUIVALENTBUILDINGS ..................................................................................................................................... 93FIGURE 8.21 FRAGILITY CURVES OF FIVE-STOREY BUILDINGS DESIGNED TO PGA=0.25G AND DC MANALYZED USING IDA METHOD: (TOP) FRAME BUILDINGS; (BOTTOM) WALL-EQUIVALENTBUILDINGS ..................................................................................................................................... 94FIGURE 8.22 FRAGILITY CURVES OF FIVE-STOREY BUILDINGS DESIGNED TO PGA=0.25G AND DC MANALYZED USING IDA METHOD: (TOP) FRAME BUILDINGS; (BOTTOM) WALL BUILDINGS ............ 95FIGURE 8.23 BEAM FRAGILITY CURVES IN YIELDING STATE FOR FIVE-STOREY FRAME-EQUIVALENTBUILDING DESIGNED TO DC M AND PGA=0.20G (LEFT) AND WALL-EQUIVALENT BUILDINGDESIGNED TO DC H AND PGA=0.25G (RIGHT).............................................................................. 96FIGURE 8.24 BEAM FRAGILITY CURVES IN YIELDING STATE FOR EIGHT-STOREY FRAME-EQUIVALENTBUILDING DESIGNED TO DC M AND PGA=0.20G (LEFT) AND WALL-EQUIVALENT BUILDINGDESIGNED TO DC M AND PGA=0.25G (RIGHT)............................................................................. 97FIGURE 8.25 BEAM FRAGILITY CURVES IN YIELDING STATE FOR FIVE-STOREY FRAME BUILDINGDESIGNED TO PGA=0.25G AND DC M (LEFT) AND DC H (RIGHT). ............................................... 978


IndexFIGURE 8.26 BEAM FRAGILITY CURVES IN ULTIMATE STATE FOR FIVE-STOREY FRAME-EQUIVALENTBUILDING DESIGNED TO DC M AND PGA=0.25G (LEFT) AND WALL-EQUIVALENT BUILDINGDESIGNED TO DC M AND PGA=0.20G (RIGHT)............................................................................. 98FIGURE 8.27 BEAM FRAGILITY CURVES IN ULTIMATE STATE FOR EIGHT-STOREY FRAME-EQUIVALENTBUILDING DESIGNED TO DC M AND PGA=0.20G (LEFT) AND WALL-EQUIVALENT BUILDINGDESIGNED TO DC M AND PGA=0.25G (RIGHT)............................................................................. 98FIGURE 8.28 BEAM FRAGILITY CURVES IN ULTIMATE STATE FOR FIVE-STOREY FRAME BUILDINGDESIGNED TO PGA=0.25G AND DC M (LEFT) AND DC H (RIGHT). ............................................... 98FIGURE 8.29 COLUMN FRAGILITY CURVES IN YIELDING STATE FOR FIVE-STOREY FRAME-EQUIVALENTBUILDING DESIGNED TO DC M AND PGA=0.25G (LEFT) AND WALL-EQUIVALENT BUILDINGDESIGNED TO DC M AND PGA=0.20G (RIGHT)............................................................................. 99FIGURE 8.30 COLUMN FRAGILITY CURVES IN YIELDING STATE FOR EIGHT-STOREY FRAME-EQUIVALENT BUILDING DESIGNED TO DC M AND PGA=0.20G (LEFT) AND WALL BUILDINGDESIGNED TO DC M AND PGA=0.20G (RIGHT)............................................................................. 99FIGURE 8.31 COLUMN FRAGILITY CURVES IN YIELDING STATE FOR FIVE-STOREY FRAME BUILDINGDESIGNED TO DC M AND PGA=0.20G AND (LEFT) PGA=0.25G (RIGHT). ................................... 100FIGURE 8.32 COLUMN FRAGILITY CURVES IN ULTIMATE STATE FOR FIVE-STOREY WALL -EQUIVALENTBUILDING DESIGNED TO DC M AND PGA=0.25G (LEFT) AND WALL BUILDING DESIGNED TO DCM AND PGA=0.25G (RIGHT)........................................................................................................ 100FIGURE 8.33 COLUMN FRAGILITY CURVES IN ULTIMATE STATE FOR FIVE-STOREY FRAME-EQUIVALENTBUILDING DESIGNED TO DC H AND PGA=0.25G (LEFT) AND WALL-EQUIVALENT BUILDINGDESIGNED TO DC H AND PGA=0.25G (RIGHT)............................................................................ 101FIGURE 8.34 COLUMN FRAGILITY CURVES IN ULTIMATE STATE FOR EIGHT-STOREY FRAME-EQUIVALENT BUILDING DESIGNED TO DC M AND PGA=0.25G (LEFT) AND WALL-EQUIVALENTBUILDING DESIGNED TO DC M AND PGA=0.25G (RIGHT). ......................................................... 101FIGURE 8.35 COLUMN FRAGILITY CURVES IN ULTIMATE STATE FOR FIVE-STOREY FRAME BUILDINGDESIGNED TO DC M AND PGA=0.20G AND (LEFT) DC H AND PGA=0.25G (RIGHT). ................. 101FIGURE 8.36 WALL FRAGILITY CURVES IN YIELDING STATE FOR FIVE-STOREY FRAME-EQUIVALENTBUILDING DESIGNED TO DC M AND PGA=0.25G (LEFT) AND WALL BUILDING DESIGNED TO DCM AND PGA=0.20G (RIGHT)........................................................................................................ 102FIGURE 8.37 WALL FRAGILITY CURVES IN YIELDING STATE FOR FIVE-STOREY FRAME-EQUIVALENTBUILDING DESIGNED TO DC M AND PGA=0.25G (LEFT) AND WALL BUILDING DESIGNED TO DCM AND PGA=0.20G (RIGHT)........................................................................................................ 1029


IndexFIGURE 8.38 WALL FRAGILITY CURVES IN ULTIMATE STATE IN FLEXURE FOR FIVE-STOREY FRAME-EQUIVALENT BUILDING DESIGNED TO DC M AND PGA=0.25G (LEFT) AND WALL-EQUIVALENTBUILDING DESIGNED TO DC H AND PGA=0.25G (RIGHT). .......................................................... 103FIGURE 8.39 WALL FRAGILITY CURVES IN ULTIMATE STATE IN FLEXURE FOR FIVE-STOREY WALLBUILDING DESIGNED TO DC M AND PGA=0.25G (LEFT) AND EIGHT-STOREY WALL BUILDINGDESIGNED TO DC M AND PGA=0.20G (RIGHT)........................................................................... 103FIGURE 8.40 BEAM FRAGILITY CURVES FOR A) YIELDING AND B) ULTIMATE STATE FOR FIVE-STOREYFRAME-EQUIVALENT BUILDING DESIGNED TO DC M AND PGA=0.25G ANALYZED WITH IDA(LEFT), SPO (MIDDLE) AND LFM (RIGHT). .................................................................................. 104FIGURE 8.41 BEAM FRAGILITY CURVES FOR A) YIELDING AND B) ULTIMATE STATE FOR FIVE-STOREYWALL-EQUIVALENT BUILDING DESIGNED TO DC M AND PGA=0.25G ANALYZED WITH IDA(LEFT), SPO (MIDDLE) AND LFM (RIGHT). .................................................................................. 104FIGURE 8.42 BEAM FRAGILITY CURVES FOR A) YIELDING AND B) ULTIMATE STATE FOR FIVE-STOREYWALL BUILDING DESIGNED TO DC M AND PGA=0.25G ANALYZED WITH IDA (LEFT), SPO(MIDDLE) AND LFM (RIGHT)........................................................................................................ 105FIGURE 8.43 BEAM FRAGILITY CURVES FOR A) YIELDING AND B) ULTIMATE STATE FOR EIGHT-STOREYFRAME-EQUIVALENT BUILDING DESIGNED TO DC M AND PGA=0.20G ANALYZED WITH IDA(LEFT), SPO (MIDDLE) AND LFM (RIGHT). .................................................................................. 105FIGURE 8.44 BEAM FRAGILITY CURVES FOR A) YIELDING AND B) ULTIMATE STATE FOR EIGHT -STOREYWALL-EQUIVALENT BUILDING DESIGNED TO DC M AND PGA=0.20G ANALYZED WITH IDA(LEFT), SPO (MIDDLE) AND LFM (RIGHT). .................................................................................. 106FIGURE 8.45 BEAM FRAGILITY CURVES FOR A) YIELDING AND B) ULTIMATE STATE FOR EIGHT -STOREYWALL BUILDING DESIGNED TO DC M AND PGA=0.20G ANALYZED WITH IDA (LEFT), SPO(MIDDLE) AND LFM (RIGHT)........................................................................................................ 106FIGURE 8.46 BEAM FRAGILITY CURVES FOR A) YIELDING AND B) ULTIMATE STATE FOR FIVE -STOREYFRAME BUILDING DESIGNED TO DC M AND PGA=0.20G ANALYZED WITH IDA (LEFT), SPO(MIDDLE) AND LFM (RIGHT)........................................................................................................ 107FIGURE 8.47 COLUMN FRAGILITY CURVES FOR C) YIELDING AND D) ULTIMATE STATE FOR FIVE-STOREY FRAME-EQUIVALENT BUILDING DESIGNED TO DC M AND PGA=0.20G ANALYZED WITHIDA (LEFT), SPO (MIDDLE) AND LFM (RIGHT). .......................................................................... 108FIGURE 8.48 COLUMN FRAGILITY CURVES FOR C) YIELDING AND D) ULTIMATE STATE FOR FIVE-STOREY WALL-EQUIVALENT BUILDING DESIGNED TO DC M AND PGA=0.25G ANALYZED WITHIDA (LEFT), SPO (MIDDLE) AND LFM (RIGHT). .......................................................................... 10810


IndexFIGURE 8.49 COLUMN FRAGILITY CURVES FOR C) YIELDING AND D) ULTIMATE STATE FOR FIVE-STOREY WALL BUILDING DESIGNED TO DC M AND PGA=0.25G ANALYZED WITH IDA (LEFT),SPO (MIDDLE) AND LFM (RIGHT). .............................................................................................. 109FIGURE 8.50 COLUMN FRAGILITY CURVES FOR C) YIELDING AND D) ULTIMATE STATE FOR EIGHT-STOREY FRAME-EQUIVALENT BUILDING DESIGNED TO DC M AND PGA=0.20G ANALYZED WITHIDA (LEFT), SPO (MIDDLE) AND LFM (RIGHT). .......................................................................... 109FIGURE 8.51 COLUMN FRAGILITY CURVES FOR C) YIELDING AND D) ULTIMATE STATE FOR EIGHT -STOREY WALL-EQUIVALENT BUILDING DESIGNED TO DC M AND PGA=0.20G ANALYZED WITHIDA (LEFT), SPO (MIDDLE) AND LFM (RIGHT). .......................................................................... 110FIGURE 8.52 COLUMN FRAGILITY CURVES FOR C) YIELDING AND D) ULTIMATE STATE FOR EIGHT -STOREY WALL BUILDING DESIGNED TO DC M AND PGA=0.20G ANALYZED WITH IDA (LEFT),SPO (MIDDLE) AND LFM (RIGHT). .............................................................................................. 110FIGURE 8.53 COLUMN FRAGILITY CURVES FOR C) YIELDING AND D) ULTIMATE STATE FOR FIVE -STOREY FRAME BUILDING DESIGNED TO DC M AND PGA=0.20G ANALYZED WITH IDA (LEFT),SPO (MIDDLE) AND LFM (RIGHT). .............................................................................................. 111FIGURE 8.54 FRAGILITY CURVES OF WALLS FOR THE ULTIMATE DAMAGE STATE IN SHEAR OF A FIVE-STOREY WALL-EQUIVALENT BUILDING DESIGNED TO DC M AND PGA=0.20G. ......................... 114FIGURE 8.55 FRAGILITY CURVES OF WALLS FOR THE ULTIMATE DAMAGE STATE IN SHEAR OF A FIVE-STOREY WALL BUILDING DESIGNED TO DC M AND PGA=0.20G. ............................................... 114FIGURE 8.56 FRAGILITY CURVES OF WALLS FOR THE ULTIMATE DAMAGE STATE IN SHEAR OF A EIGHT-STOREY WALL BUILDING DESIGNED TO DC M AND PGA=0.20G. ............................................... 11511


IndexLIST OF TABLESTABLE 3.1: MATERIAL FACTORS AND VALUES ...................................................................................... 34TABLE 4.1 BASIC VALUES OF THE BEHAVIOUR FACTOR, Q O ................................................................... 36TABLE 4.2 BASIC FACTORED VALUES OF THE BEHAVIOR FACTOR, Q O ................................................... 37TABLE 4.3 DEPTHS OF BEAMS (H B ) AND COLUMNS (H C ) FOR FIVE-STOREY FRAME BUILDINGS [ADAPTEDFROM PAPAILIA, 2011] .................................................................................................................. 38TABLE 4.4 DEPTHS OF BEAMS (H B ) AND COLUMNS (H C ) AND WALL LENGTHS (L W ) FOR WALL-FRAMEDUAL BUILDINGS [ADAPTED FROM PAPAILIA, 2011] .................................................................... 39TABLE 5.1: ACCELEROGRAM RECORDS USED IN THE ANALYSIS............................................................ 48TABLE 7.1 VALUES OF COEFFICIENT OF VARIATION FOR DM-CAPACITY VALUES ................................ 70TABLE 7.2 VALUES OF COEFFICIENT OF VARIATION FOR DM-DEMAND VALUES .................................. 70TABLE 8.1 MODAL PERIODS AND PARTICIPATING MASSES FOR FRAME SYSTEMS ................................. 72TABLE 8.2 MODAL PERIODS AND PARTICIPATING MASSES FOR FRAME-EQUIVALENT DUAL SYSTEMS . 72TABLE 8.3 MODAL PERIODS AND PARTICIPATING MASSES FOR WALL-EQUIVALENT DUAL SYSTEMS ... 73TABLE 8.4 MODAL PERIODS AND PARTICIPATING MASSES FOR WALL DUAL SYSTEMS ......................... 73TABLE 8.5 MEDIAN PGA (G) AT ATTAINMENT OF THE DAMAGE STATE IN 5-STOREY FRAME SYSTEMS 74TABLE 8.6 MEDIAN PGA (G) AT ATTAINMENT OF THE DAMAGE STATE IN 5-STOREY FRAME-EQUIVALENT SYSTEMS .................................................................................................................. 74TABLE 8.7 MEDIAN PGA (G) AT ATTAINMENT OF THE DAMAGE STATE IN 5-STOREY WALL-EQUIVALENT DUAL SYSTEMS ........................................................................................................ 75TABLE 8.8 MEDIAN PGA (G) AT ATTAINMENT OF THE DAMAGE STATE IN 5-STOREY WALL SYSTEMS . 75TABLE 8.9 MEDIAN PGA (G) AT ATTAINMENT OF THE DAMAGE STATE IN 8-STOREY FRAME-EQUIVALENT DUAL SYSTEMS ........................................................................................................ 7512


IndexTABLE 8.10 MEDIAN PGA (G) AT ATTAINMENT OF THE DAMAGE STATE IN 8-STOREY WALL-EQUIVALENT DUAL SYSTEMS ........................................................................................................ 76TABLE 8.11 MEDIAN PGA (G) AT ATTAINMENT OF THE DAMAGE STATE IN 8-STOREY WALL SYSTEMS....................................................................................................................................................... 76TABLE 8.12 MEDIAN PGA (G) AT ATTAINMENT OF THE ULTIMATE DAMAGE STATE FOR WALLS IN 5-STOREY BUILDINGS ..................................................................................................................... 112TABLE 8.13 MEDIAN PGA (G) AT ATTAINMENT OF THE ULTIMATE DAMAGE STATE FOR WALLS IN 8-STOREY BUILDINGS ..................................................................................................................... 112TABLE 8.14 MEDIAN PGA (G) AT ATTAINMENT OF THE ULTIMATE DAMAGE STATE IN SHEAR FORWALLS IN 5-STOREY BUILDINGS .................................................................................................. 113TABLE 8.15 MEDIAN PGA (G) AT ATTAINMENT OF THE ULTIMATE DAMAGE STATE IN SHEAR FORWALLS IN 8-STOREY BUILDINGS .................................................................................................. 11313


IndexLIST OF SYMBOLSA cE cdE cm(EI) b,i(EI) c,iF bF iF V,EdGH clH iH stIcIsI wK cK sL bL cl,iL sM EbM EcM Edocross section areadesign value <strong>of</strong> the modulus <strong>of</strong> elasticity <strong>of</strong> concretesecant modulus <strong>of</strong> elasticity <strong>of</strong> concreteeffective rigidity <strong>of</strong> the beams in storey ieffective rigidity <strong>of</strong> the columns in storey itotal lateral seismic shear (“base shear”)seismic horizontal force in storey itotal vertical loadpermanent (dead) loadclear height <strong>of</strong> a columntransverse storey forces which represent the effect <strong>of</strong> the inclination φistorey heightthe moment <strong>of</strong> inertia <strong>of</strong> concrete cross sectionthe second moment <strong>of</strong> area <strong>of</strong> reinforcement, about the centre <strong>of</strong> area <strong>of</strong> theconcretesecond moment <strong>of</strong> area (uncracked concrete section) <strong>of</strong> shear <strong>wall</strong>factor for effects <strong>of</strong> cracking, creep etc.factor for contribution <strong>of</strong> reinforcementbay lengthbeam clear span in storey ishear span <strong>of</strong> a memberseismic bending moment at beam endsseismic bending moment at column endsbending moment at the base <strong>of</strong> a <strong>wall</strong>, as obtained from the elastic analysis forthe design seismic action14


IndexM elM Rd,b,i-M Rd,b,j+M RdoM yNN EdPGAPGVQQ dQ EqSS aS a,dsS DS d (T)S e (T)TT 1T cT effV CD,cV EcV g+ψq,oV NV oV R,cV R,cyclV R0V RsV SV tot,baseV <strong>wall</strong>,baseXelastic seismic moment at the end <strong>of</strong> the elementdesign value <strong>of</strong> negative beam moment resistance at enddesign value <strong>of</strong> positive beam moment resistance at endflexural capacity at the base section <strong>of</strong> a <strong>wall</strong>yield momentaxial forcedesign value <strong>of</strong> the applied axial forcePeak ground accelerationPeak ground velocityimposed (live) loadload for the persistent <strong>and</strong> transient design situationCombination <strong>of</strong> actions for seismic design situationssoil factor according to EC8Spectral accelerationspectral acceleration necessary to cause the certain damage state to occurSpectral displacementDesign spectrumelastic response spectrumvibration period <strong>of</strong> a single-degree-<strong>of</strong>-freedom systemfundamental period <strong>of</strong> vibration <strong>of</strong> a buildingcorner period at the upper limit <strong>of</strong> the constant acceleration region <strong>of</strong> theelastic spectrumeffective period <strong>of</strong> vibrationcapacity-design shear <strong>of</strong> the columnsseismic shear force at column endsshear force at end regions <strong>of</strong> interior beams due to quasi-permanent gravityloadscontribution <strong>of</strong> the element axial load to its shear resistanceshear force due to gravity loadsshear force at diagonal cracking <strong>of</strong> a membershear resistance under cyclic loadingshear capacity before plastic hingingthe contribution <strong>of</strong> transverse reinforcement to shear resistanceshear dem<strong>and</strong> before plastic hingingtotal base shear <strong>of</strong> the buildingthe fraction <strong>of</strong> the building total base shear taken by the <strong>wall</strong>sr<strong>and</strong>om variable15


Indexalaα 1a cya emα ga ha ma sla vbb ib ob woc vdd 1d bLf bcf cdf ckf cmf mcf ydf ykf yLf ymf ywhh bh ch otension shifteffectiveness factor for confinement by transverse reinforcementis the value by which the horizontal seismic design action is multiplied inorder to first reach the flexural resistance in any member in the structure,while all other design actions remain constantzero-one variable for the type <strong>of</strong> loadingratio <strong>of</strong> elastic moduli (steel-to-concrete)design ground acceleration on type A ground according to EC8reduction factor for heightreduction factor for number <strong>of</strong> memberszero-one variable accounting for the slippage <strong>of</strong> longitudinal bars from theanchorage zone beyond the end sectionzero-one variablewidth <strong>of</strong> compression zonethe centreline spacing <strong>of</strong> longitudinal bars (indexed by i) laterally restrainedby a stirrup corner or a cross-tie along the perimeter <strong>of</strong> the cross-sectionwidth <strong>of</strong> confined core <strong>of</strong> a column or in the boundary element <strong>of</strong> a <strong>wall</strong><strong>wall</strong> web thicknesscoefficient <strong>of</strong> variationeffective depth <strong>of</strong> a sectiondistance <strong>of</strong> the center <strong>of</strong> the compression reinforcement from the extremecompression fibresmean tension bar diameternormalised compressive strength <strong>of</strong> the masonry unitsdesign value <strong>of</strong> concrete compressive strengthcharacteristic value <strong>of</strong> concrete compressive strengthmean value <strong>of</strong> concrete compressive strengthspecified compressive strength <strong>of</strong> the mortardesign value <strong>of</strong> steel yield strengthcharacteristic value <strong>of</strong> steel yield strengthyield stress <strong>of</strong> the longitudinal barsmean value <strong>of</strong> steel yield strengthyield stress <strong>of</strong> transverse steeldepth <strong>of</strong> a cross sectionbeam depthcolumn depthdepth <strong>of</strong> confined core <strong>of</strong> a column or in the boundary element <strong>of</strong> a <strong>wall</strong>16


Indexh wi g<strong>wall</strong> heightradius <strong>of</strong> gyration <strong>of</strong> the uncracked concrete sectionk 1 ; k 2 relative flexibilities or rotational restrains at member ends 1 <strong>and</strong> 2ll 0l wmmaxV i,d,bm effm inn stn flxqq osx yzz iΔδ iΘΣM Rd,bΣM Rd,cα uββ Dβ Rβ Sβ Spγ cγ gγ qclear height <strong>of</strong> compression member between end restrainseffective length <strong>of</strong> a member<strong>wall</strong> lengthmean <strong>of</strong> the non-logarithmized variables <strong>of</strong> a lognormal distributioncapacity design shear at the end regions <strong>of</strong> interior beamseffective mass <strong>of</strong> a buildingmass <strong>of</strong> floor irelative normal force for the design value <strong>of</strong> the applied axial forcenumber <strong>of</strong> storeysnumber <strong>of</strong> flexible <strong>frame</strong>s per one stiffbehaviour factorbasic values <strong>of</strong> the behaviour factorst<strong>and</strong>ard deviation <strong>of</strong> the non-logarithmized variables <strong>of</strong> a lognormaldistributionneutral axis depth at flexural yieldinglength <strong>of</strong> the internal lever arm <strong>of</strong> a memberthe height <strong>of</strong> the mass, , above the level <strong>of</strong> application <strong>of</strong> the seismic action(foundation or top <strong>of</strong> a rigid basement)interstorey drift from mid-height <strong>of</strong> the storey i to the mid-height i+1 <strong>of</strong> the<strong>frame</strong>rotation <strong>of</strong> restraining member for bending moment Msum <strong>of</strong> beam design flexural capacitiessum <strong>of</strong> column design flexural capacitiesthe value by which the horizontal seismic design action is multiplied in orderto form plastic hinges in a number <strong>of</strong> sections sufficient for the development<strong>of</strong> overall structural instability, while all other design actions remain constantthe normalised composite log-normal st<strong>and</strong>ard deviationlower bound factor for the horizontal design spectrumdispersion <strong>of</strong> the capacity (in terms <strong>of</strong> st<strong>and</strong>ard deviation)dispersion <strong>of</strong> the dem<strong>and</strong> (in terms <strong>of</strong> st<strong>and</strong>ard deviation)dispersion <strong>of</strong> the spectral value (in terms <strong>of</strong> st<strong>and</strong>ard deviation)partial factor for concretepartial factor for permanent actionpartial factor for variable action17


Indexγ Rdγ sδ iεε RVε sV,elε sζε uε yεζζ sζ uζ umζ yζ ymκ 1κ 2λμplμ ζμ φνξξ yππ 1π 2π dπ sπ wπ νδφ 0factor accounting for steel strain hardeningpartial factor for steelthe displacement <strong>of</strong> floor from an elastic analysis <strong>of</strong> the structure for the set <strong>of</strong>lateral forcescapacity design magnification factoruncertainty factor for shear capacitydem<strong>and</strong> uncertainty factor for shear failure (prior to the formation <strong>of</strong> a plastichinge)uncertainty factor <strong>of</strong> the chord rotation dem<strong>and</strong>capacity uncertainty factoruncertainty factor for the yielding chord rotationdamping correction factor with a reference <strong>of</strong> for 5% viscous dampingmember chord rotationmean chord rotation dem<strong>and</strong>ultimate chord rotationthe expected chord rotation capacitychord rotation at yieldingthe expected chord rotation value at yieldingfactor which depends on concrete strength classfactor which depends on axial force <strong>and</strong> slendernessslenderness rationormal distribution meanratio <strong>of</strong> the plastic part <strong>of</strong> the rotation dem<strong>and</strong> at the end <strong>of</strong> the member to thevalue at yieldingcurvature ductility factoraxial load ratio, positive for compressionreduction factor for unfavourable permanent actionsneutral axis depth at yieldinggeometric reinforcement ratioratio <strong>of</strong> the tension reinforcementratio <strong>of</strong> the compression reinforcementsteel ratio <strong>of</strong> diagonal reinforcement in each diagonal directionratio <strong>of</strong> transverse steel parallel to the loading directionthe transverse reinforcement ratioratio <strong>of</strong> “web” reinforcementnormal distribution st<strong>and</strong>ard deviationbasic value <strong>of</strong> the inclination taking account for the geometric imperfections18


Indexφ effφ iφ yψ 2ψ οω 1ω 2effective creep ratio <strong>of</strong> concreteinclination taking account for the geometric imperfectionsyield curvaturefactor for quasi-permanent value <strong>of</strong> a variable actionfactor for combination value <strong>of</strong> a variable actionmechanical reinforcement ratio <strong>of</strong> tension <strong>and</strong> “web” longitudinalreinforcementmechanical reinforcement ratio <strong>of</strong> compression longitudinal reinforcement19


Introduction1. INTRODUCTIONThis study deals with the seismic <strong>fragility</strong> <strong>of</strong> members for <strong>frame</strong> <strong>and</strong> <strong>wall</strong>-<strong>frame</strong> <strong>buildings</strong><strong>designed</strong> in accordance to EN-Eurocode 2 <strong>and</strong> 8. Prototype plan- <strong>and</strong> height-wise very regular<strong>buildings</strong> are studied. Parameters include the number <strong>of</strong> storeys, the level <strong>of</strong> Eurocode 8design (in terms <strong>of</strong> design peak ground acceleration <strong>and</strong> ductility class) <strong>and</strong> for <strong>wall</strong>-<strong>frame</strong><strong>dual</strong> systems the percentage <strong>of</strong> seismic base shear taken by the <strong>wall</strong>s.The <strong>fragility</strong> curves relate seismic ground motion to structural damage which is important inorder to denote the damage probability <strong>of</strong> the members in a structure. Fragility curves areimportant for estimating the risk from potential earthquakes <strong>and</strong> for predicting the economicalimpact for future earthquakes. They can be used for emergency response <strong>and</strong> disasterplanning by national agencies <strong>and</strong> by insurance companies for estimating the overall loss afteran earthquake event. Fragility curves can be used to mitigate risk by improving the seismiccodes.Fragility curves are constructed for generic members for each building assuming a lognormaldistribution. The probability <strong>of</strong> exceedance <strong>of</strong> each limit state is computed from theprobability distributions <strong>of</strong> the damage measures (conditional on intensity measure) <strong>and</strong> <strong>of</strong> thecorresponding capacities. The intensity measure (IM) used for the construction <strong>of</strong> the <strong>fragility</strong>curves is the peak ground acceleration (PGA) <strong>and</strong> the damage measures are the peak chordrotation <strong>and</strong> the peak shear force dem<strong>and</strong>s at member ends. <strong>Seismic</strong> performance is addressedon two damage states; the yielding <strong>and</strong> the ultimate deformation in bending or shear. Theestimations for the peak response quantities <strong>and</strong> capacities for each member are according toEurocode 8 – Part 3 [CEN, 2005].The <strong>fragility</strong> curves are developed by using the analysis results obtained from threedimensionalstructural models <strong>of</strong> the full <strong>buildings</strong> using nonlinear incremental dynamicanalysis (IDA) <strong>and</strong> nonlinear static (pushover) analysis (SPO). IDA is carried out usingfourteen semi-artificial spectrum-compatible ground motion records scaled in order to cover arange <strong>of</strong> ground motion intensities. SPO is carried out using the inverted triangulardistribution pattern <strong>and</strong> the N2 method [Fajfar et. al., 2000] is being employed to combine theresults <strong>of</strong> the static pushover analysis with the response spectrum analysis <strong>of</strong> an equivalentsingle degree-<strong>of</strong>-freedom system to compute the IM for each step <strong>of</strong> the analysis.20


IntroductionDispersions used for the construction <strong>of</strong> <strong>fragility</strong> curves from IDA take into account explicitlymodel uncertainties for the estimation <strong>of</strong> the damage measure dem<strong>and</strong>s taken from theanalysis. Estimates for the dispersions <strong>of</strong> the damage measure dem<strong>and</strong>s for the SPO methodare taken from previous studies. Both methods use estimates for the damage measurecapacities based on previous studies.The results <strong>of</strong> a simplified method using the lateral force method (LFM) taken from Papailia[2011] is compared against the results from SPO <strong>and</strong> IDA. The LFM is performed by usingsimplified models under the assumption that all beam ends in a storey have the same elasticseismic moments <strong>and</strong> inelastic chord rotation dem<strong>and</strong>s. Vertical elements are considered tohave negligible bending moments due to gravity loads <strong>and</strong> the axial force variation due toseismic action is neglected in interior columns. The shear force dem<strong>and</strong>s taken from the LFMare amplified to take into account higher mode effects.Discussion will focus on the differences between geometric <strong>and</strong> design parameters <strong>of</strong> the<strong>buildings</strong> <strong>and</strong> the differences between the alternative analysis methods. The <strong>wall</strong>s <strong>of</strong> <strong>buildings</strong><strong>designed</strong> according to Eurocode 8 for Medium Ductility Class is an important point <strong>of</strong> thediscussion since according to the results using the lateral force method they fail in shearbefore their design PGA.21


Chapter 2: Definitions <strong>and</strong> Background2. DEFINITIONS AND BACKGROUNDA brief introduction for various definitions <strong>and</strong> a review <strong>of</strong> previous studies is found in thisChapter.2.1. Building codesThe analysis, design <strong>and</strong> assessment <strong>of</strong> the <strong>buildings</strong> were performed in accordance to theEuropean St<strong>and</strong>ards; Eurocode 2 [CEN, 2004a], Eurocode 8 - Part 1 [CEN, 2004b] <strong>and</strong> Part 3[CEN, 2005]. Eurocode 2 <strong>and</strong> Eurocode 8 – Part 1 were published by the EuropeanCommittee for St<strong>and</strong>ardization (CEN) in December <strong>of</strong> 2004. Eurocode 2 is for the design <strong>of</strong>concrete structures <strong>and</strong> Eurocode 8 – Part 1 is for the seismic design <strong>of</strong> new <strong>buildings</strong>.Eurocode 8 – Part 3 was published by CEN in June 2005 for the seismic retr<strong>of</strong>it <strong>and</strong>assessment <strong>of</strong> structures. Since March 2010 all CEN member countries use the EN-Eurocodes.2.2. Performance-based requirementsPerformance-based earthquake engineering allows for design to meet more than oneperformance level thus replacing the traditional design against collapse. The performancelevel is the condition <strong>of</strong> the facility or structure after a seismic event. The seismic event isidentified by the annual probability <strong>of</strong> exceedence known as the “seismic hazard level”.In EN-Eurocodes the performance levels are associated to the Limit States <strong>of</strong> the structure.The Ultimate Limit State concerns the safety <strong>of</strong> people <strong>and</strong> the Serviceability Limit Stateconcerns the comfort <strong>of</strong> its occupants <strong>and</strong> the function <strong>and</strong> use <strong>of</strong> the structure. According toEurocode 8 – Part 1 [CEN, 2004] the following two Limit States (or performance levels) areconsidered:221. “No-(local)- collapse”: It is considered as the Ultimate Limit State. This limit stateprotects life against rare seismic events by preventing the collapse <strong>of</strong> structuralmembers. The seismic action associated with this limit state is the “design seismicaction” having 10% probability <strong>of</strong> being exceeded in 50 years (mean return period <strong>of</strong>475 years).2. “Damage Limitation”. It is considered as the Serviceability Limit State, where thestructural or non structural damage is limited under frequent seismic events. Thestructure is expected not to have any permanent deformations <strong>and</strong> should retain its


Chapter 2: Definitions <strong>and</strong> Backgroundstrength <strong>and</strong> stiffness. The seismic action associated with this limit state is the“damage limitation seismic action” with 10% probability <strong>of</strong> being exceeded in 10years (mean return period <strong>of</strong> 95 years).Eurocode 8 – Part 3 [CEN, 2005] for the assessment <strong>and</strong> retr<strong>of</strong>itting <strong>of</strong> structures has fullyadopted the performance-based approach for three performance levels:1. “Damage Limitation” (DL), structural elements are not significantly yielded <strong>and</strong> retaintheir strength <strong>and</strong> stiffness <strong>and</strong> the structure has negligible permanent drifts <strong>and</strong> norepairs are required. It is recommended that the performance objective should bereached for a 20% probability <strong>of</strong> exceedence in 50 years (return period <strong>of</strong> 225 years).2. “Significant Damage” (SD), which corresponds to the “no-(local)-collapse” accordingto EC8-Part 1, where the structure is significantly damaged but retains some resi<strong>dual</strong>lateral strength <strong>and</strong> stiffness <strong>and</strong> its vertical load bearing capacity. Non-structuralcomponents are damaged <strong>and</strong> moderate drifts are present. The structure will be able tosurvive aftershocks <strong>of</strong> moderate intensity. It is recommended that the performanceobjective should be reached for a 10% probability <strong>of</strong> exceedence in 50 years (returnperiod <strong>of</strong> 475 years).3. “Near Collapse” (NC), the structure is heavily damaged with large permanent drifts<strong>and</strong> little resi<strong>dual</strong> lateral strength or stiffness is retained although the vertical elementsare still able to retain vertical loads. The structure would most probably not be able tosurvive another earthquake. It is recommended that the performance objective shouldbe reached for a 2% probability <strong>of</strong> exceedence in 50 years (return period <strong>of</strong> 2475years).This study is addressed on two limit states; the yielding <strong>and</strong> the ultimate. The yieldingcorresponds to the “Damage Limitation” limit state <strong>and</strong> the ultimate corresponds to the “NearCollapse” limit state as defined by Eurocode 8 – Part 3 [CEN, 2005].2.3. Intensity MeasureAn Intensity Measure (IM) is the ground motion parameter that is being used in order to relatethe ground motion to the damage <strong>of</strong> the building. The selected parameter should be able tocorrelate the ground motion to the damage <strong>of</strong> the <strong>buildings</strong>. Intensity measures can be dividedinto instrumental IM <strong>and</strong> non-instrumental IM.For non-instrumental IM, macroseismic data are used in computing the empiricalvulnerability <strong>of</strong> structures. Macroseismic data is expressed in different macroseismic intensityscales, which identify the effects <strong>of</strong> ground motion, <strong>and</strong> is taken from observation <strong>of</strong> damagedue to earthquake ground motion <strong>and</strong> its effects on the earth’s surface, people <strong>and</strong> structures.Macroseismic intensity scale is a qualitative scale expressed in terms <strong>of</strong> Roman numeralsrepresenting different intensity levels. An advantage <strong>of</strong> this type <strong>of</strong> intensity measure is that itis directly related to the vulnerability <strong>of</strong> the <strong>buildings</strong> <strong>and</strong> there is no requirement to takeinstrumental measurements. The gathered data depends on the area where it is collected <strong>and</strong>how far away this area is from the epicenter.23


Chapter 2: Definitions <strong>and</strong> BackgroundThe most important IMs for non-instrumental seismicity are the MSK: Medvedev-Sponheur-Karnik Intensity scale [Medvedev <strong>and</strong> Sponheuer, 1969], the MMI: Modified MercalliIntensity Scale [Wood <strong>and</strong> Neumann, 1931], the European Macroseismic Sclae (EMS98)[Grünthal, 1998] <strong>and</strong> the MCS: Mercalli – Cancani – Sieberg [Sieberg, 1923]. The MCS wasproposed as the development <strong>of</strong> the Mercalli scale <strong>and</strong> includes twelve degrees from I“Instrumental” to XII “Cataclysmic”. MMI scale is composed <strong>of</strong> twelve degrees. MSK goesfrom I “No perceptible” to XII “Very catastrophic”.Previous studies made use <strong>of</strong> the non-instrumental intensity measures using the empiricalvulnerability procedures to produce post-earthquake damage statistics [Calvi et al., 2006].Such studies include Braga et al. [1982] where the damage probability matrices have beendeveloped based on damage data obtained from the Irpinia 1980 earthquake. The <strong>buildings</strong>were separated in three classes <strong>and</strong> the matrices were based on the MSK scale for each class.Di Pasquale et al. [2005] updated Braga’s study <strong>and</strong> changed the MSK scale to the MCS scalebecause the Italian seismic catalogue is based on this intensity measure. Dolce et. al. [2003]have adapted the damage probability matrices with an additional vulnerability class using theEMS98 scale, which takes into account the <strong>buildings</strong> constructed after 1980. Singhal <strong>and</strong>Kiremidjian [1996] developed <strong>fragility</strong> curves <strong>and</strong> damage probability matrices using theModified Mercalli Intensity.In instrumental intensity measures, instruments are used in order to record the ground motion<strong>and</strong> then recorded accelerograms are processed to get the appropriate measurement. Theinstrumental intensity measures include the Peak ground Velocity (PGV), the Peak GroundAcceleration (PGA), the Peak Ground Displacement (PGD), the Spectral Acceleration at thefirst mode <strong>of</strong> vibration S a (T 1 ,5%) <strong>and</strong> the spectral displacement S d . PGV correlates well withthe earthquake magnitude <strong>and</strong> gives useful information on the ground-motion frequencycontent <strong>and</strong> strong-motion duration which influence the seismic dem<strong>and</strong>s <strong>of</strong> the structure[Akkar <strong>and</strong> Őzen, 2006]. The Spectral Acceleration at the first mode <strong>of</strong> vibration S a (T 1 ) is<strong>of</strong>ten used since it is well suited for structures that are sensitive to the strength <strong>of</strong> thefrequency content near its first mode frequency [Vamvatsikos <strong>and</strong> Cornell, 2002].These instrumental intensity measures were used in reference studies such as Kircil <strong>and</strong> Polat[2006] where elastic pseudo-spectral acceleration was considered as an intensity measure indeveloping <strong>fragility</strong> curves for <strong>RC</strong> <strong>frame</strong> <strong>buildings</strong>. Akkar et al. [2005] constructed <strong>fragility</strong>functions for <strong>RC</strong> <strong>buildings</strong> using PGV as the IM since maximum inelastic displacements arebetter correlated with PGV than with PGA <strong>and</strong> PGV has a good correlation with MMI forlarge amplitude earthquakes. Borzi et al. [2006] used PGA as the intensity measure for thevulnerability analysis <strong>of</strong> <strong>RC</strong> <strong>buildings</strong>. PGA was used since it is consistent with the parameterused in seismic hazard maps in the current codes.More complicated IMs have been introduced such as the vector-valued IMs by Baker [2005]which consists <strong>of</strong> two parameters; the spectral acceleration <strong>and</strong> epsilon. Epsilon is found to beable to predict the structural response. It is defined as the difference between spectralacceleration <strong>of</strong> a record <strong>and</strong> the mean <strong>of</strong> the ground motion prediction equation at a given24


Chapter 2: Definitions <strong>and</strong> Backgroundperiod. Neglecting the effect <strong>of</strong> epsilon gives conservative estimates on the response <strong>of</strong> thestructure.The ground motion IM that is being used in this study is the Peak Ground Acceleration(PGA). The reason for this choice is due to the simplicity <strong>of</strong> its use <strong>and</strong> due to the fact that theresults can be easily compared against the design acceleration <strong>of</strong> the structures.2.4. Damage measuresDamage measure (DM) is a scalar quantity that can be deducted from the analysis <strong>and</strong>characterizes the response <strong>of</strong> the structural model due to seismic loading. Selecting a suitableDM depends on the application <strong>and</strong> the structure.The damage measures for members that are used in reference studies include:25 The peak chord rotation dem<strong>and</strong> at member end The peak shear force dem<strong>and</strong> The local Park <strong>and</strong> Ang Damage Index [1985]. The node rotations Displacement ductility, μThe Park <strong>and</strong> Ang Damage index takes into account the damage due to maximumdeformation <strong>and</strong> the damage due to repeated cycles <strong>of</strong> inelastic deformation. Thedisplacement ductility is associated with the inelastic response <strong>and</strong> is defined as the ratio <strong>of</strong>the maximum displacement to the yield displacement.Common damage measures selected for the assessment <strong>of</strong> <strong>buildings</strong> as a whole include: The resi<strong>dual</strong> deformation The global Park <strong>and</strong> Ang Damage Index [1985] Maximum base shear The peak ro<strong>of</strong> drift Interstorey drift ratio The peak interstorey drift angle , θ max = max(θ 1 … … θ n ) Peak floor accelerationsThe peak interstorey drift angle is used for structural damage <strong>of</strong> <strong>buildings</strong> <strong>and</strong> relates well tojoint rotations. The peak floor accelerations are used for damage to non-structural componentsin multi-storey <strong>buildings</strong>. [Vamvatsikos <strong>and</strong> Cornell, 2002]. The Interstorey drift ratio is theratio <strong>of</strong> the maximum storey displacement over the storey height. It gives significantinformation on the structural <strong>and</strong> non-structural damage.Examples <strong>of</strong> reference studies that used the DMs above include Singhal <strong>and</strong> Kiremidjian[1996], where the global damage index based on Park <strong>and</strong> Ang [1985], in order to develop<strong>fragility</strong> curves <strong>and</strong> damage probability matrices for <strong>RC</strong> <strong>frame</strong> structures. Őzer <strong>and</strong> Erberik[2008] developed <strong>fragility</strong> curves for the damage measure <strong>of</strong> the maximum interstorey drift


Chapter 2: Definitions <strong>and</strong> Backgroundratio <strong>and</strong> a s<strong>of</strong>tening index (SI) which was originally proposed by DiPasquale <strong>and</strong> Cakmak[1987]. SI takes a value according to the stiffness change due to inelastic action. In anotherreference study, Borzi et al. [2006] based the building limit conditions on displacementswhich are well correlated with building damage.For the purposes <strong>of</strong> this study the damage measures used are the peak chord rotations at amember end <strong>and</strong> the peak shear force dem<strong>and</strong>s. The chord rotation at a member end is definedas the angle between the tangent to the member section there <strong>and</strong> the chord connecting thetwo members ends as shown in Figure 2.1. When plastic hinge forms in the member end, thechord rotation is equal to the plastic hinge rotation.Figure 2.1 Definition <strong>of</strong> chord rotation [adapted from Fardis, 2009]2.5. <strong>Seismic</strong> Vulnerability Assessment MethodologiesDifferent methodologies for the seismic vulnerability assessment <strong>of</strong> <strong>buildings</strong> are usedaccording to the data available <strong>and</strong> the uncertainties considered. These methods include theempirical, expert opinion, analytical <strong>and</strong> hybrid methods.262.5.1. Empirical Fragility CurvesEmpirical methods for the vulnerability assessment <strong>of</strong> <strong>buildings</strong> are based on the damageobserved after a seismic event. The two main types <strong>of</strong> empirical methods are the damageprobability matrices (DPM) <strong>and</strong> the continuous vulnerability functions. DPM is a form <strong>of</strong>conditional probability <strong>of</strong> obtaining a damage level due to the IM. The continuousvulnerability functions illustrate the probability <strong>of</strong> exceeding a given damage state as afunction <strong>of</strong> the seismic IM. The advantages <strong>of</strong> using empirical fragilities are that the observeddamage from the earthquakes is the most realistic way to model <strong>fragility</strong> <strong>and</strong> takes intoaccount many uncertainties such as soil-structure-interaction <strong>and</strong> variability <strong>of</strong> the structural


Chapter 2: Definitions <strong>and</strong> Backgroundcapacity. The disadvantages are that the empirical vulnerability functions require that thesurvey forms are not incomplete <strong>and</strong> the way post-processing is done with the data should notbe deficient. These curves need to be derived for <strong>buildings</strong> in the same region <strong>and</strong> shouldaccount for damage subjected after a specific earthquake event. Often undamaged <strong>buildings</strong>are not recorded so when deriving the vulnerability analysis it is difficult to assess the totalnumber <strong>of</strong> <strong>buildings</strong> in the analysis [SYNER-G, 2012]. Empirical vulnerability cannot modelthe evaluation <strong>of</strong> retr<strong>of</strong>it options <strong>and</strong> do not cover all building types <strong>and</strong> values <strong>of</strong> IM. [Calviet al. 2006].Sabetta et al. [1998] developed vulnerability curves from post earthquake damage surveys <strong>and</strong>estimated ground motion. The damage surveys <strong>of</strong> nearly 50000 <strong>buildings</strong> after earthquakeevents in Italy together with estimates <strong>of</strong> strong ground motion parameters from attenuationrelationships was used for the development <strong>of</strong> <strong>fragility</strong> curves. The binomial distribution <strong>of</strong>the damage was plotted as a function <strong>of</strong> PGA, Arias Intensity <strong>and</strong> Effective Peak Accelerationfor three structural classes <strong>and</strong> six damage levels according to the MSK macroseismic scale.Effective Peak Acceleration is defined as the mean response spectral acceleration divided by afactor <strong>of</strong> 2.5.Sarab<strong>and</strong>i et. al. [2004] developed empirical <strong>fragility</strong> functions from recent earthquakes withdata taken from the Northridge, California earthquake in 1994 <strong>and</strong> the Chi-Chi earthquake in1999 in Taiwan. Buildings situated near the strong motion recording stations were used in theassessment <strong>and</strong> were divided into two groups according to their distance from the recordingstation. Empirical <strong>fragility</strong> curves are produced for steel moment <strong>frame</strong>s, concrete <strong>frame</strong>s,concrete shear <strong>wall</strong>s, wood <strong>frame</strong> <strong>and</strong> unreinforced masonry <strong>buildings</strong>.Rota et al. [2006] developed typological <strong>fragility</strong> curves from post-earthquake survey data onthe damage observed on the <strong>buildings</strong> after Italian earthquakes from the past three decades.150,000 survey building records have been post processed to define the empirical damageprobability matrices for different building typologies. Typological <strong>fragility</strong> curves have beenobtained using advanced nonlinear regression methods. Typological risk maps were thendeveloped for both single damage state <strong>and</strong> for average loss parameters after combining thehazard definitions, <strong>fragility</strong> curves <strong>and</strong> inventory data.2.5.2. Expert Opinion methodExert opinion method is a method to construct <strong>fragility</strong> curves based on the judgment <strong>and</strong>information taken by experts. The probability <strong>of</strong> damage for different building typologiescovering a range <strong>of</strong> ground motion intensities are taken from the opinion <strong>of</strong> experts. Theadvantage <strong>of</strong> the method is that it is not affected by the quantity <strong>and</strong> quality <strong>of</strong> the structuraldamage data <strong>and</strong> statistics. The main disadvantage is that the method is restricted on theknowledge <strong>and</strong> experience <strong>of</strong> the experts consulted. The study <strong>of</strong> Kostov et a. [2007]produced damage probability matrices for <strong>buildings</strong> in S<strong>of</strong>ia according to the EMS-98. Thedamage probability matrices were then converted in vulnerability curves.27


Chapter 2: Definitions <strong>and</strong> Background2.5.3. Analytical Fragility CurvesThis method features a more detailed vulnerability assessment with direct physical meaning.The analytical <strong>fragility</strong> curves are computed by constructing appropriate structural modelswhich express the probability <strong>of</strong> damage computed under increasing seismic intensity. Figure2.2 summarizes the basic procedures that are being followed in order to calculate theanalytical vulnerability curves or damage probability matrices. The advantage <strong>of</strong> this methodis that it provides results that are very close to reality. One <strong>of</strong> the main disadvantages <strong>of</strong>analytical vulnerability curves is that they are computationally dem<strong>and</strong>ing <strong>and</strong> timeconsuming. Also the capability <strong>of</strong> modelling the structure significantly affects the reliability<strong>of</strong> the results.Eurocode 8 - Part 3 [CEN, 2005] provides guidelines for the assessment <strong>of</strong> existing <strong>buildings</strong>which may be used to develop analytical <strong>fragility</strong> curves. The methods <strong>of</strong> analysis include thelateral force analysis, the modal response spectrum analysis, the nonlinear static pushoveranalysis, the nonlinear time-history dynamic analysis. The nonlinear static method appliesforces to the model which includes the nonlinear properties <strong>of</strong> the elements. The nonlineardynamic analysis although time consuming gives results that are closer to reality. Also itallows the influence <strong>of</strong> the variability <strong>of</strong> the accelerogram to be taken into account. Thesemethods are performed in order to compute the seismic action effects.In order to choose the type <strong>of</strong> analysis to be performed <strong>and</strong> the appropriate confidence factorvalues EC8 - Part 3 defines three knowledge levels:KL1: Limited KnowledgeKL2: Normal KnowledgeKL3: Full KnowledgeThe factors that determine the knowledge levels are the geometrical properties <strong>of</strong> thestructural system <strong>and</strong> non structural elements, the details (regarding the reinforcement inreinforced concrete members, the connections between steel members, the floor diaphragmconnection to lateral resisting structure etc.) <strong>and</strong> the mechanical properties <strong>of</strong> the constituentmaterials used.For the purpose <strong>of</strong> this study analytical <strong>fragility</strong> curves have been developed using nonlineartime-history dynamic analysis <strong>and</strong> nonlinear static (pushover) analysis. The <strong>buildings</strong>assessed belong to the Full knowledge level (KL3) <strong>of</strong> Eurocode 8 – Part3 since allgeometrical properties, details <strong>and</strong> mechanical properties <strong>of</strong> the materials are known.28


Chapter 2: Definitions <strong>and</strong> BackgroundFigure 2.2 Flowchart to describe the components <strong>of</strong> the calculation <strong>of</strong> analytical vulnerability curve[adapted from Dumova-Jovanoska (2004)]Existing studies for the computation <strong>of</strong> seismic <strong>fragility</strong> curves for <strong>RC</strong> <strong>buildings</strong> that arebased on the analytical method include the following.Singhal <strong>and</strong> Kiremidjian [1996] developed <strong>fragility</strong> curves <strong>and</strong> damage probability matricesusing Monte Carlo simulation for low-rise, mid-rise <strong>and</strong> high-rise <strong>RC</strong> <strong>frame</strong>s using Park <strong>and</strong>Ang (1985) damage index to identify different degrees <strong>of</strong> damage. The analysis was based onnonlinear dynamic analysis where the ground motion is characterized by spectral acceleration.For the computation <strong>of</strong> damage probability matrices the modified Mercalli intensity was usedas the ground motion parameter.B. Borzi et. al. [2006] use analytical methods where the nonlinear behavior <strong>of</strong> a r<strong>and</strong>ompopulation <strong>of</strong> <strong>RC</strong> <strong>buildings</strong> was defined with simplified pushover <strong>and</strong> displacement basedprocedures. The vulnerability curves were generated by comparing the displacementcapacities by the pushover analysis with the displacement dem<strong>and</strong>s obtained from responsespectrum <strong>of</strong> each building in the r<strong>and</strong>om population. The vulnerability curves wereformulated using the conditional probability <strong>of</strong> exceeding a certain damage limit state in terms<strong>of</strong> the IM.Dumova et.al [2000] evaluated the vulnerability curves/ damage probability matrices usinganalytical methods for <strong>frame</strong>-<strong>wall</strong> <strong>RC</strong> <strong>buildings</strong> <strong>designed</strong> according to the Macedonian design29


Chapter 2: Definitions <strong>and</strong> Backgroundcode. Two sets <strong>of</strong> <strong>buildings</strong> were analyzed; six storey <strong>frame</strong> <strong>buildings</strong> <strong>and</strong> sixteen storey<strong>frame</strong>-<strong>wall</strong> <strong>buildings</strong>. Nonlinear time-history analysis was performed for a set <strong>of</strong> synthetictime histories <strong>and</strong> the response <strong>of</strong> the structure to the earthquake excitation was definedaccording to modified Park <strong>and</strong> Ang (1985) damage model using five damage states toexpress the condition <strong>of</strong> damage. The probability <strong>of</strong> occurrence <strong>of</strong> damage was assumed to benormal probabilistic distribution.Masi [2003] employed analytical methods for the seismic vulnerability assessment <strong>of</strong> existing<strong>RC</strong> <strong>frame</strong> <strong>buildings</strong> (bare, regularly infilled <strong>and</strong> pilotis) <strong>designed</strong> only to gravity loads for<strong>buildings</strong> representative <strong>of</strong> the Italian building block <strong>of</strong> the past 30 years <strong>designed</strong> accordingto the building codes at the period <strong>of</strong> their construction. The analysis was performed usingnonlinear time-history analysis using artificial <strong>and</strong> natural accelerograms. The vulnerabilitywas characterized through the use <strong>of</strong> European Macroseismic Scale.Kirçil <strong>and</strong> Polat [2006] evaluated the behavior <strong>of</strong> mid-rise <strong>RC</strong> <strong>frame</strong> <strong>buildings</strong> usinganalytical methods. The building stock represented <strong>buildings</strong> <strong>of</strong> 3, 5 <strong>and</strong> 7 storeys that were<strong>designed</strong> according to the (1975) Turkish seismic code. In this study only yielding <strong>and</strong>collapse damage levels are considered <strong>and</strong> they were determined analytically under the effect<strong>of</strong> twelve artificial accelerograms using incremental dynamic analysis. The yielding <strong>and</strong>collapse capacities are evaluated by statistical methods to develop <strong>fragility</strong> curves in terms <strong>of</strong>elastic pseudo-spectral acceleration. Lognormal distribution is assumed for the construction <strong>of</strong>the <strong>fragility</strong> curves.302.5.4. Hybrid methodsHybrid damage probability matrices <strong>and</strong> vulnerability functions combine damage observedafter earthquakes with damage obtained from analytical methods. This method isadvantageous when there is lack <strong>of</strong> observational data. Also post-earthquake damage data canbe used to calibrate the analytical model. Observational data can reduce the computationaleffort that would normally be required to perform complete analytical analysis.Kappos et. al. [1998] developed the damage probability matrices using a hybrid procedurewhere data from past earthquakes was combined with results <strong>of</strong> nonlinear dynamic analysisfor typical Greek <strong>buildings</strong> <strong>designed</strong> for the 1959 codes. The results <strong>of</strong> the dynamic analysiswere used in order to obtain a global damage index <strong>and</strong> correlated with loss in terms <strong>of</strong> cost <strong>of</strong>repair. Observational damage from the 1978 Thessaloniki earthquake was combined with theanalytical damage results.2.6. <strong>Seismic</strong> safety assessment <strong>of</strong> <strong>RC</strong> <strong>buildings</strong> <strong>designed</strong> to EC8The efficacy <strong>of</strong> Eurocode 8 <strong>and</strong> design provisions <strong>and</strong> the expected performance has beenevaluated in the past. The following studies were performed for the seismic safety assessment<strong>of</strong> <strong>RC</strong> <strong>buildings</strong>.Panagiotakos <strong>and</strong> Fardis [2004] evaluated the performance <strong>of</strong> <strong>RC</strong> <strong>buildings</strong> <strong>designed</strong>according to Eurocode 8 using nonlinear analysis. <strong>RC</strong> <strong>frame</strong>s <strong>of</strong> 4, 8 <strong>and</strong> 12 storeys were


Chapter 2: Definitions <strong>and</strong> Background<strong>designed</strong> for a PGA <strong>of</strong> 0.2g or 0.4g <strong>and</strong> to the three ductility classes. The limit states areconsidered as in EC8 for the life-safety (475 years) <strong>and</strong> the damage limitation (95 years) <strong>and</strong>are evaluated through nonlinear seismic response analysis. It was found that the design toDuctility Class High (DC H) or Medium (DC M) is more cost effective than DC Low even inmoderate seismicity <strong>and</strong> more cost effective than the 2000 Greek national codes. It was als<strong>of</strong>ound that the large differences in material quantities <strong>and</strong> detailing <strong>of</strong> the alternative designsdo not translate into large differences in performance.Rivera <strong>and</strong> Petrini [2011] investigate the efficacy <strong>of</strong> the Eurocode 8 force-based designprovisions for <strong>RC</strong> <strong>frame</strong>s. This study evaluates whether the <strong>RC</strong> <strong>buildings</strong> that are <strong>designed</strong>according to the EC8 provisions have the expected performance. Four, eight <strong>and</strong> sixteenstorey <strong>RC</strong> <strong>frame</strong> <strong>buildings</strong> were <strong>designed</strong> <strong>and</strong> analyzed using the EC8 response spectrumanalysis. Nonlinear time-history analysis was performed to determine the seismic response <strong>of</strong>the structures <strong>and</strong> validate the EC8 forced base designs. The results indicate that the design <strong>of</strong>flexural members in medium-to-long period structures is not significantly influenced by thechoice <strong>of</strong> effective member stiffness. However the interstorey drift dem<strong>and</strong>s calculated aresignificantly affected. Design storey forces <strong>and</strong> interstorey drift dem<strong>and</strong>s found using thecode’s force base procedure varied substantially from the results <strong>of</strong> the nonlinear time-historyanalysis. From the results it was concluded that EC8 may yield life-safe designs. Also theseismic performance <strong>of</strong> <strong>RC</strong> <strong>frame</strong> <strong>buildings</strong> <strong>of</strong> the same type <strong>and</strong> ductility class can be highlynon-uniform.Rutenberg <strong>and</strong> Nsieri [2005] evaluated the seismic shear dem<strong>and</strong> in ductile cantilever <strong>wall</strong>systems. Two aspects were considered; (1) Single <strong>wall</strong>s or a system <strong>of</strong> equal-length <strong>wall</strong>s <strong>and</strong>(2) resisting system consisting <strong>of</strong> <strong>wall</strong>s <strong>of</strong> different length. The results <strong>of</strong> the parametricstudies showed that DC M <strong>and</strong> DC H <strong>wall</strong>s <strong>designed</strong> to EC8 provisions are in need <strong>of</strong> revisionsince for DC M <strong>wall</strong>s the inelastic amplification which takes into account the higher modeeffects as required in EC8 is under-conservative whereas the amplification used for DC H<strong>wall</strong>s according to the detailed procedure per Keintzel [1990] overestimates the shear dem<strong>and</strong>in <strong>wall</strong>s for most cases..31


Chapter 3: Description <strong>of</strong> Buildings3. DESCRIPTION OF BUILDINGSFor the scope <strong>of</strong> this study pure <strong>frame</strong> <strong>and</strong> <strong>wall</strong>-<strong>frame</strong> (<strong>dual</strong>) reinforced concrete <strong>buildings</strong>were analyzed <strong>and</strong> assessed. Two analysis methods were performed: nonlinear static <strong>and</strong>nonlinear dynamic analysis <strong>of</strong> the structures comprising different design <strong>and</strong> geometricparameters. The parameters, methods <strong>and</strong> assumptions made when modelling the structuresare explained <strong>and</strong> discussed in this section.3.1. Typology <strong>of</strong> <strong>buildings</strong>The design <strong>and</strong> detailing <strong>of</strong> the <strong>frame</strong> <strong>and</strong> the <strong>wall</strong>-<strong>frame</strong> (<strong>dual</strong>) <strong>buildings</strong> correspond tocertain design parameters including:Number <strong>of</strong> storeys: 5 <strong>and</strong> 8 storeys<strong>Seismic</strong> Design level per EC8 for Ductility classo Medium Ductility Class (DC M)o High Ductility Class (DC H)<strong>Seismic</strong> Design level per EC8 for design PGAo 0.20go 0.25gFor <strong>wall</strong>-<strong>frame</strong> <strong>dual</strong> <strong>buildings</strong>, the fraction <strong>of</strong> the seismic base shears taken by the <strong>wall</strong>s: Frame-equivalent <strong>dual</strong> system 0.35V tot,base ≤ V <strong>wall</strong>,base ≤ 0.50V tot,base Wall-equivalent <strong>dual</strong> system 0.50V tot,base ≤V <strong>wall</strong>,base ≤ 0.65V tot,base Wall system V <strong>wall</strong>,base ≥ 0.65V tot,base3.2. Geometry <strong>of</strong> <strong>buildings</strong>The <strong>buildings</strong> are regular in plan <strong>and</strong> in elevation having storey height <strong>of</strong> H st =3.0m, where allstoreys are <strong>of</strong> the same height. The <strong>buildings</strong> consist <strong>of</strong> five bays along the two horizontaldirections <strong>of</strong> bay length L b =5.0m with the same bay length throughout the plan.The <strong>buildings</strong> consist <strong>of</strong> square columns, beams <strong>of</strong> width 0.3m <strong>and</strong> slab thickness <strong>of</strong> 150mm.The size <strong>of</strong> columns is constant throughout all storeys <strong>and</strong> the size <strong>of</strong> beams is constantthroughout each storey. The perimeter beams <strong>and</strong> exterior columns have half the elasticrigidity <strong>of</strong> interior ones <strong>and</strong> corner columns have one quarter <strong>of</strong> elastic rigidity <strong>of</strong> interiorones.32


Chapter 3: Description <strong>of</strong> BuildingsIn <strong>wall</strong>-<strong>frame</strong>d <strong>dual</strong> systems two <strong>wall</strong>s on each direction are placed as shown in Figure 3.1<strong>and</strong> Figure 3.3 sharing the same displacements with the <strong>frame</strong>. The geometry <strong>of</strong> the <strong>frame</strong>building is illustrated in Figure 3.2 for a five- <strong>and</strong> eight-storey building. The beam <strong>and</strong>column depths <strong>and</strong> <strong>wall</strong> lengths for <strong>wall</strong>-<strong>frame</strong> <strong>buildings</strong> are shown in Table 4.4 <strong>and</strong> the beam<strong>and</strong> column depths for <strong>frame</strong> <strong>buildings</strong> are shown in Table 4.3.Figure 3.1 Plan <strong>of</strong> <strong>wall</strong>-<strong>frame</strong> (<strong>dual</strong>) <strong>buildings</strong> [Papailia, 2011]Figure 3.2 Geometry <strong>of</strong> <strong>frame</strong> <strong>buildings</strong> [Papailia, 2011]33


Chapter 3: Description <strong>of</strong> BuildingsFigure 3.3 Structural 3D model taken from ANSRuop for five – storey <strong>dual</strong> system3.3. MaterialsThe material strengths <strong>and</strong> partial factors are taken according to Annex C <strong>of</strong> Eurocode 2[CEN,2004a]. The structural materials consist <strong>of</strong> concrete <strong>of</strong> class C25/30, having a nominalstrength <strong>of</strong> 25MPa <strong>and</strong> Tempocore steel <strong>of</strong> grade S500 (Class C). The following tableprovides the material properties for steel <strong>and</strong> concrete <strong>and</strong> their partial factors.Table 3.1: Material factors <strong>and</strong> valuesPartial factorsPartial factor for Concrete c 1.5Partial factor for Steel s 1.15ConcreteC25/30Concrete compressive strength f ck 25 MPaDesign compressive strength f cd = cc f ck / c 16.67MPaMean concrete compressive strength f cm =f ck +8MPa 33MPaMean axial tensile concrete strength f ctm 2.56 MPaSecant modulus <strong>of</strong> elastic <strong>of</strong> concrete E cm 30470 MPaDesign value <strong>of</strong> modulus <strong>of</strong> elasticity E cd =E cm / cE 25392 MPaConcrete Cover c nom 30mmSteelS500Characteristic yield strength <strong>of</strong> reinforcement f yk 500MPaDesign yield strength <strong>of</strong> reinforcement f yd = f yk / s 434.78MPaMean yield strength <strong>of</strong> reinforcement f ym =1.15 f yk 575 MPaDesign value <strong>of</strong> modulus <strong>of</strong> elasticity <strong>of</strong> steel E s 200000 MPaFor the seismic vulnerability assessment the mean values for material strengths are being used(f ym =575MPa for reinforcing steel <strong>and</strong> f cm =33MPa for concrete).34


Chapter 4: Design <strong>of</strong> Buildings4. DESIGN OF BUILDINGS4.1. Actions on structure <strong>and</strong> assumptionsThe actions considered in the analysis correspond to the seismic design situation <strong>and</strong> thepersistent <strong>and</strong> transient design situation according to EN1990.The combination <strong>of</strong> vertical actions for the seismic design situation is:Where,Q EQ = G + ψ 2 Q ( 4.1)ψ 2 quasi-permanent value <strong>of</strong> a variable action factor (=0.3)G permanent load (=7 kN/m 2 )Q imposed load (=2 kN/m 2 )The combination for the persistent <strong>and</strong> transient design situation according to EN1990 isgiven by :where:Q d =max(ξγ g G+ γ g Q ; γ g G+ ψ o γ g Q) ( 4.2)ξ is the reduction factor for unfavourable permanent actions (=0.85)ψ 0 is the factor for combination value <strong>of</strong> a variable action (=0.7)γ g is the partial factor for permanent action (=1.35)γ q is the partial factor for variable action (=1.5)The permanent load acting on the structure is 7kN/m 2 , which includes the weight <strong>of</strong> the slab,finishing, partitions <strong>and</strong> facades <strong>and</strong> the weight <strong>of</strong> the beams, columns <strong>and</strong> <strong>wall</strong>s. Theoccupancy loads (live loads) amount to 2kN/m 2 .35


Chapter 4: Design <strong>of</strong> BuildingsThe design <strong>of</strong> the building was taken from Papailia [2011] where the “lateral force method” isused to proceed with the design according to EC8 [CEN,2004b]. In order to compute the baseshear force, as required by the lateral force method, the design spectrum <strong>and</strong> the fundamentalperiod is used. The design spectrum is computed by the use <strong>of</strong> the behaviour factor q obtainedas explained in the section below <strong>and</strong> the fundamental period <strong>of</strong> the structure is obtained bythe Rayleigh quotient.In concrete <strong>buildings</strong> the stiffness <strong>of</strong> the load bearing elements are evaluated by taking intoaccount the effects <strong>of</strong> cracking. The cracking effect corresponds to the yielding initiation <strong>of</strong>the reinforcement. In Eurocode 8 [CEN, 2004], this simplification can be taken into accountby assuming that the flexural <strong>and</strong> shear stiffness properties are one half <strong>of</strong> the initialuncracked stiffness <strong>of</strong> the element.4.2. Behaviour factors <strong>and</strong> local ductilityIn force-based design according to EC8 [CEN,2004b], the use <strong>of</strong> the behaviour factoraccounts for a simplification in design where the forces found by elastic analysis are reduced.The values <strong>of</strong> the basic behaviour factor for <strong>buildings</strong> <strong>designed</strong> to DC M <strong>and</strong> DC H are givenin Table 4.1 for <strong>frame</strong> systems, <strong>wall</strong>-<strong>frame</strong> systems <strong>and</strong> uncoupled <strong>wall</strong> systems. Uncoupled<strong>wall</strong> systems are defined as <strong>wall</strong> systems which are linked by a connecting medium which isnot effective in flexure.Table 4.1 Basic values <strong>of</strong> the behaviour factor, q oDC MDC HFrame system, <strong>wall</strong>-<strong>frame</strong> system 3.0 α u /α 1 4.5 α u /α 1Uncoupled Wall system 3.0 4.0 α u /α 1Where,α 1 the value by which the horizontal seismic design action is multiplied to reach theflexural resistance in any member in the structure while other design actions remain constant.α u the value by which the horizontal seismic design action is multiplied to form plastichinges in a number <strong>of</strong> sections sufficient for the development <strong>of</strong> structural instability, whileall other design actions remain constant.The ratio <strong>of</strong> α u /α 1 for <strong>frame</strong> or <strong>frame</strong>-equivalent <strong>dual</strong> system may be taken equal to 1.3, for<strong>wall</strong>-equivalent systems equal to 1.2 <strong>and</strong> for <strong>wall</strong> system with two uncoupled <strong>wall</strong>s perhorizontal direction equal to 1.0. Thus the basic values <strong>of</strong> the behaviour factor, q o , are:36


Chapter 4: Design <strong>of</strong> BuildingsTable 4.2 Basic factored values <strong>of</strong> the behavior factor, q oFrame–equivalent / Frame systems Wall-equivalent Wall systemsDC M 3.9 3.6 3.0DC H 5.85 5.4 4.04.3. Design procedureThis section describes the procedure that was followed for the sizing <strong>of</strong> beams, columns <strong>and</strong><strong>wall</strong>s.4.3.1. Sizing <strong>of</strong> beams <strong>and</strong> columns in <strong>frame</strong> systemsThe sizing <strong>of</strong> beams <strong>and</strong> columns in <strong>frame</strong> systems was performed according to Eurocode 8[CEN,2004b] <strong>and</strong> Eurocode 2 [CEN,2004a]. The sizing <strong>of</strong> the beams <strong>and</strong> the columns wastaken from Papailia [2011]. The procedure to size the member is described in this section.Eurocode 2 [CEN,2004a] gives a simplified criterion for the slenderness ratio <strong>of</strong> isolatedcolumns:λ = l oi g≤ λ lim = 20 A B Cn( 4.3)Where,i gl 0is the radius <strong>of</strong> gyration <strong>of</strong> the uncracked concrete sectionis the effective lengthn Is the normalised axial force taken as n=N ed / A c f cd <strong>and</strong> N ed is the design value <strong>of</strong> theapplied axial force.The default values for A, B <strong>and</strong> C are A=0.7, B=1.1 <strong>and</strong> C=0.7.The effective length is given by:Where,l o = H cl . max 1 + 10 k 1k 2k 1 +k 2; 1 + k 11+k 11 + k 21+k 2( 4.4)k i is the column rotational stiffness at the end node i relative to the total restrainingstiffness <strong>of</strong> the members framing in the plane <strong>of</strong> bending.k i = θ iM iEI c,effH cl=EI c,effH cl( 4.5)4 EI c,eff+4 EI b ,effH cl L cl37


Chapter 4: Design <strong>of</strong> BuildingsWhere,L clis the clear length <strong>of</strong> a beam framing into node iEI b,eff is the cracked flexural rigidity, taking into account creepEI c,eff = E s I s + E cd f ck (MPa )20K 2 I c1+φ eff( 4.6)E s <strong>and</strong> I s are the elastic modulus <strong>and</strong> the moment <strong>of</strong> inertia <strong>of</strong> the sections reinforcement withrespect to the centroid <strong>of</strong> the section. I c is the moment <strong>of</strong> inertia <strong>of</strong> the uncracked grossconcrete section <strong>and</strong> K 2 is :K 2 = nλ= 1 N Ed l o≤ 0.20 ( 4.7)170 170 A c f cd i cThe effective length <strong>of</strong> the column <strong>and</strong> the size <strong>of</strong> the section are both unknown at thebeginning, thus iterations are performed after dimensioning <strong>of</strong> the top beam reinforcement atthe supports.In pure <strong>frame</strong> systems the depths <strong>of</strong> the columns <strong>and</strong> beams are chosen iteratively as theminimum values meeting the requirements <strong>of</strong> Eurocode 2 [CEN,2004a] <strong>and</strong> Eurocode 8[CEN,2004b]. This takes into account the above implementation for the slenderness limit tomeet the negligible second order effects <strong>and</strong> the 0.5% storey drift limit per EC8 under thedamage limitation seismic action, where the 50% <strong>of</strong> the design seismic action is taken.In the following table the sizes <strong>of</strong> the beams <strong>and</strong> columns are presented for different designparameters (ductility class <strong>and</strong> design PGA)Table 4.3 Depths <strong>of</strong> beams (h b ) <strong>and</strong> columns (h c ) for five-storey <strong>frame</strong> <strong>buildings</strong> [adapted from Papailia,DesignPGADC2011]h b (m)h c (m)0.20g M/H 0.40 0.550.25g M/H 0.45 0.554.3.2. Sizing <strong>of</strong> beams, columns <strong>and</strong> <strong>wall</strong>s in <strong>wall</strong>-<strong>frame</strong> (<strong>dual</strong>) systemsIn <strong>dual</strong> (<strong>wall</strong>-<strong>frame</strong>) <strong>buildings</strong> the lateral force procedure according to EC8 [CEN,2004b] wasperformed <strong>and</strong> iterated until certain criteria were met. The sizing <strong>of</strong> the members is takenfrom Papailia [2011]. The depths <strong>of</strong> columns (h c ) <strong>and</strong> beams (h b ) <strong>and</strong> the length <strong>of</strong> the <strong>wall</strong>s(l w ) were chosen iteratively to meet the following requirements according to EC8[CEN,2004b]:38


Chapter 4: Design <strong>of</strong> BuildingsMeet the storey drift ratio <strong>of</strong> 0.5% according to Eurocode 8 [CEN, 2004b].To cover the three cases for the requirements <strong>of</strong> the <strong>wall</strong> to total base shear fractionfollowing the different behavior factors <strong>and</strong> design rules per EC8:o Frame-equivalent <strong>dual</strong> system 0.35V tot,base ≤ V <strong>wall</strong>,base ≤ 0.50V tot,baseo Wall-equivalent <strong>dual</strong> system0.50V tot,base ≤V <strong>wall</strong>,base ≤ 0.65V tot,baseo Wall systemV <strong>wall</strong>,base ≥ 0.65V tot,baseIn the following table the sizes <strong>of</strong> the beams <strong>and</strong> columns <strong>and</strong> the length <strong>of</strong> the <strong>wall</strong>s arepresented for different design parameters (<strong>wall</strong> base shear fraction, ductility class <strong>and</strong> designPGA)Table 4.4 Depths <strong>of</strong> beams (h b ) <strong>and</strong> columns (h c ) <strong>and</strong> <strong>wall</strong> lengths (l w ) for <strong>wall</strong>-<strong>frame</strong> <strong>dual</strong> <strong>buildings</strong>[adapted from Papailia, 2011]Design DC 5 storeys 8 storeysPGA h b (m) h c (m) l w (m) V <strong>wall</strong>,b (%) h b (m) h c (m) l w (m) V <strong>wall</strong>,b (%)0.20g M/H a 0.40 0.40 1.5 37 0.45 0.45 2.0/- 42/-2.0 53 3.0/3.0 b 63/732.5 65 4.0/- 76/-0.25g M/H a 0.45 0.45 2.0 44 0.50 0.45 2.0/- 40/-2.5 57 3.0/- 61/-3.5/3.5 b 73/81 4.0/5.5 b 74/90a When DC M <strong>and</strong> DC H have different fraction <strong>of</strong> base shear <strong>and</strong> <strong>wall</strong> length, this is distinguishedwith a slash, where the left h<strong>and</strong> side is the DC M <strong>and</strong> the right h<strong>and</strong> side the DC H.b Wall width is 0.5m. In all other cases <strong>wall</strong> width is 0.25m.4.4. Dimensioning <strong>of</strong> BeamsThe longitudinal reinforcement for ULS in bending in beams is <strong>designed</strong> for the persistent<strong>and</strong>-transient<strong>and</strong> the seismic design situations using the lateral force method. Thereinforcement in the effective beam flange was taken to be 500mm 2 .For the seismic design situation, the dimensioning <strong>of</strong> the end regions <strong>of</strong> the beams is done inaccordance to the capacity design rules computed using the design base shears at the memberends, according to EC8 [CEN,2004b]. The beam design shear forces were determined underthe transverse load through the seismic design situation <strong>and</strong> the end moments, M i,d , whichcorrespond to the formation <strong>of</strong> plastic hinges.The end moments M i,d depend on the moment resistances <strong>of</strong> the columns it is connected to<strong>and</strong> the moment resistance <strong>of</strong> the beams itself. It can be found using:M i,d = γ Rd M Rb ,i min(1,M RcM Rb) ( 4.8)39


Chapter 4: Design <strong>of</strong> BuildingsWhere,γ Rd factor accounting for steel strain hardening, equal to 1.0 <strong>of</strong> DC M <strong>and</strong> 1.2 for DC H.M Rb,iΣM Rcdesign value <strong>of</strong> the beam moment resistance at end isum <strong>of</strong> the column design moment <strong>of</strong> resistance.ΣM Rb sum <strong>of</strong> the beam design moment <strong>of</strong> resistance, framing to the point.Thus the capacity design shear at the member ends corresponds to:Where,V Ed,i = M 1,d +M 2,dl cl+ V g+ψq ,0 ( 4.9)V Ed,icapacity design shear at the member ends.V g+ψq ,0 Shear force at the end regions due to the transverse quasi-permanent loadsunder the design seismic situation.Figure 4.1Capacity design values <strong>of</strong> shear forces on beams [CEN, 2004]4.5. Dimensioning <strong>of</strong> ColumnsThe vertical reinforcement <strong>of</strong> the columns for the ULS in bending was <strong>designed</strong> for the axialload taken from the actions <strong>of</strong> the seismic design situation. The detailing rules according toEurocode 8 [CEN, 2004] are taken into account for each seismic design level.The dimensioning for the end regions <strong>of</strong> the columns is computed in accordance to thecapacity design rule through the design shear forces. The design shear forces are based on theelement equilibrium under the end moments M i,d which correspond to the formation <strong>of</strong> plastichinges as shown in Figure 4.2. The end moments are computed by taking into account themoment resistances <strong>of</strong> the beams to which it is connected <strong>and</strong> the moment resistances <strong>of</strong> thecolumn itself.40


Chapter 4: Design <strong>of</strong> BuildingsThe end moments M i,d are determined through:M i,d = γ Rd M Rc,i min(1,M RbM Rc) ( 4.10)Where,γ Rd factor accounting for steel strain hardening <strong>and</strong> the confinement <strong>of</strong> the concrete <strong>of</strong> thecompression zone <strong>of</strong> the section, equal to 1.1.M Rc,iΣM Rcdesign value <strong>of</strong> the column moment resistance at end isum <strong>of</strong> the column design moment <strong>of</strong> resistance.ΣM Rb sum <strong>of</strong> the beam design moment <strong>of</strong> resistance, framing to the point.Thus the capacity design shear at the member ends corresponds to:Where,V Ed,i = M 1,d +M 2,dH cl( 4.11)V Ed,iH clcapacity design shear at the end regions.clear height <strong>of</strong> column.41


Chapter 4: Design <strong>of</strong> BuildingsFigure 4.2 Capacity design shear force in columns [CEN 2004]4.6. Dimensioning <strong>of</strong> WallsThe design shear force <strong>and</strong> moments for the <strong>wall</strong>s are according to the capacity designprinciples <strong>and</strong> their calculation is explained below according to EC8 [CEN,2004b]. Thevalues for the axial force are computed from the analysis <strong>of</strong> the structure in the seismic designsituation using the lateral force method.The design bending moment diagram along the height <strong>of</strong> slender <strong>wall</strong>s should be given by anenvelope <strong>of</strong> the bending moment diagram from analysis, with a tension drift, as shown inFigure 4.3. Slender <strong>wall</strong>s are defined as <strong>wall</strong>s having a height to length ratio greater than 2.0.The envelope is assumed to be linear since there are no discontinuities over the height <strong>of</strong> thebuilding. It takes into account potential development <strong>of</strong> moments due to higher mode inelasticresponse after the formation <strong>of</strong> plastic hinge at the bottom <strong>of</strong> the <strong>wall</strong>, thus the region abovethis critical height is <strong>designed</strong> to remain elastic.42


Chapter 4: Design <strong>of</strong> BuildingsKEY:aba 1moment diagram from analysisdesign envelopetension driftFigure 4.3: Design envelope for bending moments in the slender <strong>wall</strong>s (left: <strong>wall</strong> systems ; right: <strong>dual</strong>systems ) [CEN 2004]The design envelope <strong>of</strong> shear forces, as shown in Figure 4.4, takes into account theuncertainties <strong>of</strong> higher modes. The flexural capacity at the base <strong>of</strong> the <strong>wall</strong> M Rd exceeds theseismic design bending moment derived from the analysis, M Ed . Thus the design shear found′for the analysis, V Ed , is magnified by the magnification factor i.e. the ratio <strong>of</strong> M Rd /M Ed . Themagnification factor depends on the ductility class <strong>of</strong> the structure. The design base shear isthus computed by:Where,′V Ed = V Ed ( 4.12) For <strong>wall</strong>s in DC M <strong>buildings</strong> the magnification factor, is taken as 1.5 For <strong>wall</strong>s in DC H <strong>buildings</strong> the magnification factor, is taken as:Where,ε = q . γ Rdqγ Rd overstrength factor taken as 1.2.M RdM Ed2+ 0.1S e (T c )S e (T 1 )S e (T 1 ) ordinate <strong>of</strong> the elastic response spectrum at fundamental periodS e (T C ) ordinate <strong>of</strong> the elastic response spectrum at corner period2≤ q ( 4.13 )43


Chapter 4: Design <strong>of</strong> BuildingsKEY:abcAshear diagram from analysismagnified shear diagramdesign envelopeV <strong>wall</strong>,baseB V <strong>wall</strong>,top ≥V <strong>wall</strong>,base /2Figure 4.4 Design envelope <strong>of</strong> the shear forces in the <strong>wall</strong>s <strong>of</strong> a <strong>dual</strong> system [CEN 2004]At the critical regions <strong>of</strong> the <strong>wall</strong> the curvature ductility factor μ φ is required in order tocalculate the confining reinforcement within boundary elements. The curvature ductilityfactor is now the product <strong>of</strong> the basic behaviour factor q o found in Section 4.2 <strong>and</strong> the ratio <strong>of</strong>the design bending moment from the analysis M Ed , to the design flexural resistance M Rd . Thisconfining reinforcement should extend vertically up to a height h cr <strong>of</strong> the critical region <strong>and</strong>horizontally along the length l c <strong>of</strong> the boundary element.The length <strong>of</strong> this boundary element is the measure from extreme compression fibre to thepoint where spalling occurs in concrete due to large compressive strains. As a minimum theboundary region should be taken as being larger than 0.15.l w or 1.5.b w . The <strong>wall</strong> criticalregion height, h cr , is estimated using the following relationship:h cr =max l w , h w6≤2l wH cl for : n st ≤62 H cl for : n st≤7( 4.14)44


Chapter 4: Design <strong>of</strong> BuildingsWhere,n sth wH cll wthe number <strong>of</strong> storeysthe <strong>wall</strong> heightis the clear storey height. The base is defined as the level <strong>of</strong> the foundation or the top<strong>of</strong> the basement storey.is the length <strong>of</strong> the cross section <strong>of</strong> the <strong>wall</strong>Above the height <strong>of</strong> the critical region, h cr , the rules <strong>of</strong> EN1992 apply for the dimensioning <strong>of</strong>vertical <strong>and</strong> horizontal reinforcement.45


Chapter 5: Structural modelling <strong>and</strong> analysis methods5. ANALYSIS METHODS AND MODELLING ASSUMPTIONSFor the construction <strong>of</strong> the <strong>fragility</strong> curves different analysis methods were performed eachfollowing different modelling assumptions. For the purpose <strong>of</strong> this study two methods wereperformed; the nonlinear static pushover analysis <strong>and</strong> the nonlinear dynamic analysis. Theresults from these methods were then compared against a simplified method following thelateral force analysis method by Papailia [2011]. The following section explains the procedure<strong>and</strong> assumptions for the analysis methods <strong>and</strong> structural models.5.1. Nonlinear Static “Pushover” Analysis“Static pushover” (SPO) analysis is performed for the evaluation <strong>of</strong> the <strong>buildings</strong> according toEurocode 8 – Part 1 [CEN,2004b]. SPO is performed using the structural model assumptionsdetermined in Chapter 5.3 <strong>and</strong> using the computational s<strong>of</strong>tware <strong>of</strong> ANSRuop.SPO is essentially an extension <strong>of</strong> the “lateral force method” <strong>of</strong> static analysis, but in thenonlinear regime. This method simulates the inertial forces due to a horizontal component <strong>of</strong>the seismic action. These lateral forces F i increase throughout the analysis <strong>and</strong> are applied insmall steps on the mass m i in proportion to the pattern <strong>of</strong> horizontal displacements, Φ i . Themagnitude <strong>of</strong> the lateral loads is controlled by a <strong>and</strong> magnified in each step.46F i = a m i Φ i (5.1)According to EC8 [CEN,2004b], pushover analysis can be performed using the “modalpattern” which simulates the inertial forces <strong>of</strong> the first mode shape in the elastic regime. Sincethe <strong>buildings</strong> in the current study meet the conditions <strong>of</strong> the linear static analysis an “invertedtriangular” lateral load pattern is applied. In this method the horizontal displacements Φ i aresuch that Φ i = z i , where z i is the height <strong>of</strong> the mass m i above the level <strong>of</strong> the application <strong>of</strong>the seismic action.The N2 method is employed according Fajfar et. al. [2000] as adopted in EC8 [CEN,2004b].This method combines the pushover analysis <strong>of</strong> the multi-degree-<strong>of</strong>-freedom (MDOF) modelwith the response spectrum analysis <strong>of</strong> an equivalent single-degree-<strong>of</strong>-freedom (SDOF)system. This method is formulated in the acceleration – displacement format thus it enablesthe visualization <strong>of</strong> the relations between various quantities controlling the seismic response.Thus using this method the ground accelerations at the top <strong>of</strong> the soil are related to seismic


Chapter 5: Structural modelling <strong>and</strong> analysis methodsdem<strong>and</strong>s for every step <strong>of</strong> the analysis. The dem<strong>and</strong>s are then compared against the limitstates according to Eurocode 8 – Part 3 [CEN, 2005], therefore the PGA value that causesyielding <strong>and</strong> ultimate chord rotations <strong>and</strong> the ultimate shear force for each member on thestructure is computed. Also the damage indices (ratio <strong>of</strong> the damage measure dem<strong>and</strong> to thedamage measure capacity for a member) can be easily obtained for every step <strong>of</strong> the analysis<strong>and</strong> used to construct <strong>fragility</strong> curves.5.2. Incremental Dynamic AnalysisIncremental dynamic analysis (IDA) is a method by Vamvatsikos <strong>and</strong> Cornell [2002] whereseismic dem<strong>and</strong>s are estimated accurately through a series <strong>of</strong> nonlinear time-history analysesusing several ground motion records scaled to multiple levels <strong>of</strong> intensity. IDA is used inorder to uncover the structural model’s behavior in the elastic phase, the yielding <strong>and</strong> thenonlinear inelastic phase. The damage measures that are <strong>of</strong> interest are the peak chord rotationdem<strong>and</strong>s <strong>and</strong> the shear force dem<strong>and</strong>s at member ends. IDA is performed using the structuralmodel assumptions determined in Chapter 5.3 <strong>and</strong> using the computational s<strong>of</strong>tware <strong>of</strong>ANSRuop.As defined by Vamvatsikos <strong>and</strong> Cornell [2002], the scale factor (SF) is the scalar λ used inorder to uniformly scale up or down the amplitude <strong>of</strong> the accelerogram. The accelerogramsare scaled by a scalable Intensity Measure (IM) (i.e. excitation PGA).Where,α λ = λ . α 1 (5.2)α λα 1λis the scaled accelerogram time-history recordis the unscaled accelerogram time-history recordis the scale factorThe records were scaled so that they cover a range <strong>of</strong> PGA values which range from 0.05g to0.95g with a step <strong>of</strong> 0.05g. The total number <strong>of</strong> analyses performed for each building sums upto 266 having 14 analyses for each <strong>of</strong> the 19 selected IM points.Eurocode 8-Part 3 [CEN, 2005] requires at least seven nonlinear dynamic analyses <strong>and</strong> thenthe average response quantities from these analyses are used as the damage measure dam<strong>and</strong>s.For this study 14 records have been selected as shown in Table 5.1 <strong>and</strong> Figure 5.2 in order totake into account the differences in the characteristics <strong>of</strong> the ground motion. Seven historicearthquakes were used to get semi-artificial bidirectional ground motion records for twohorizontal directions X <strong>and</strong> Y. Each accelerogram is modified to be compatible with a smooth5%- damped elastic response spectrum. The spectrum consists <strong>of</strong> an acceleration sensitivepart for the periods <strong>of</strong> 0.2 to 0.6 sec, a velocity controlled part from 0.6 to 2 sec <strong>and</strong> adisplacement control part from 2 <strong>and</strong> beyond. The pseudo-acceleration spectra for the 1447


Chapter 5: Structural modelling <strong>and</strong> analysis methodsaccelerogram records are compared to the smooth 5%-damped elastic spectrum for a PGA <strong>of</strong>1g as shown in Figure 5.1.The damping matrix C is taken to be <strong>of</strong> Rayleigh type where C=a o M+a 1 K. a o <strong>and</strong> a 1 are themass <strong>and</strong> stiffness proportional damping coefficients respectively. These are obtained usingthe modal periods <strong>of</strong> the first <strong>and</strong> the second periods <strong>of</strong> the structure with the highestparticipating mass in the horizontal direction. A damping ratio <strong>of</strong> 5% is used <strong>and</strong> thus with theuse <strong>of</strong> Rayleigh damping the viscous damping ratio is lower than 5% between the range <strong>of</strong> ω 1<strong>and</strong> ω 2 <strong>and</strong> higher outside this range.The numerical integration <strong>of</strong> the equation <strong>of</strong> motion was performed using the Newmarkmethod <strong>and</strong> the Newton-Rapson algorithm for the solution algorithm for the nonlinearanalysis problem.Table 5.1: Accelerogram records used in the analysisNo Event Station Component1 Imperial Valley, 1979 BondsCorner 1402 Imperial Valley, 1979 BondsCorner 2303 Loma Prieta, 1989 Capitola 0004 Loma Prieta, 1989 Capitola 0905 Kalamata, 1986 Kalamata X6 Kalamata, 1986 Kalamata Y7 Montenegro, 1979 Herceg Novi X8 Montenegro, 1979 Herceg Novi Y9 Friuli, 1976 Tolmezzo X10 Friuli, 1976 Tolmezzo Y11 Montenegro, 1979 Ulcinj (2) X12 Montenegro, 1979 Ulcinj (2) Y13 Imperial Valley, 1940 Elcentro Array #9 18014 Imperial Valley, 1940 Elcentro Array #9 27048


Chapter 5: Structural modelling <strong>and</strong> analysis methods.Figure 5.1 Pseudo-acceleration spectra for the semi-artificial input motions compared to the smooth targetspectrum (shown with thick black line)49


Chapter 5: Structural modelling <strong>and</strong> analysis methodsFigure 5.2 Time-histories <strong>of</strong> accelerograms used in the analysis50


Chapter 5: Structural modelling <strong>and</strong> analysis methods5.3. Structural modelling for IDA <strong>and</strong> SPOANSRuop is the computational tool that is used in order to perform the modelling, seismicresponse analysis <strong>and</strong> evaluation <strong>of</strong> the structures [Kosmopoulos et al., 2005]. It is animproved <strong>and</strong> exp<strong>and</strong>ed version <strong>of</strong> ANSR-I which was developed at UC Berkeley [Mondkaretal., 1975]. The s<strong>of</strong>tware is used for the analysis <strong>of</strong> reinforced concrete structures <strong>and</strong> consists<strong>of</strong> a user interface where the user can perform the various tasks. ANSRuop was used toperform nonlinear time-history analysis <strong>and</strong> nonlinear static pushover analysis. This sectionwill explain the modelling assumptions taken for the members <strong>and</strong> the structure.Key points <strong>of</strong> the modelling <strong>of</strong> the reinforced concrete members are:For the modelling <strong>of</strong> all the reinforced concrete members inelasticity is lumped at theends. For monotonic loading the reinforced concrete members follow a bilinear Moment –curvature envelope <strong>and</strong> for the cyclic loading the members follow the Takeda hystereticrules [Takeda et. al., 1970], modified to Litton [1975] <strong>and</strong> Otani [1974]. The chordrotations <strong>and</strong> moments are calculated in accordance to the EC8 [CEN,2004b], taking intoaccount the confinement <strong>of</strong> the members.Figure 5.3 Takeda model modified by Litton <strong>and</strong> OtaniElement elastic stiffness is taken as equal to the secant stiffness at yielding (EI eff ). In orderto find this value the shear span at the yielding end <strong>of</strong> the element is required. The shearspan <strong>of</strong> the columns <strong>and</strong> the beams is taken as half the clear length between the beam-tocolumnjoints within the plane <strong>of</strong> bending. In positive or negative bending it is the averagesecant-to-yielding stiffness at the two end sections. For <strong>wall</strong>s the secant-to-yieldingstiffness <strong>of</strong> the bottom section is used with a shear span ratio <strong>of</strong> one-half the height fromthe bottom <strong>of</strong> the section to the top <strong>of</strong> the <strong>wall</strong> in the building.The <strong>wall</strong>s are modelled as cantilever <strong>wall</strong>s. Axial load acts on the <strong>wall</strong>s due to its selfweight<strong>and</strong> the floor loads. No mass is assigned due to its self-weight since it is taken intoaccount by the mass taken from the floor loads.Masses for beams <strong>and</strong> columns are lumped at the nearest node <strong>of</strong> the element <strong>and</strong> aretaken from the action <strong>of</strong> the permanent <strong>and</strong> imposed loads acting uniformly on the floors.No self-weight is assigned to the <strong>frame</strong> since it is taken into account in the floor loads.51


Chapter 5: Structural modelling <strong>and</strong> analysis methodsKey points <strong>of</strong> the modelling <strong>of</strong> the structure are:The perimeter beam <strong>and</strong> exterior columns are modelled such that they have half the elasticrigidity <strong>of</strong> interior ones. Thus both interior <strong>and</strong> exterior beam <strong>and</strong> columns have the sameseismic chord rotations dem<strong>and</strong>s whereas perimeter beam <strong>and</strong> exterior columns have halfthe elastic seismic moments <strong>of</strong> interior ones. Corner columns have one-quarter <strong>of</strong> elasticrigidity <strong>of</strong> interior ones thus the corner columns have one quarter <strong>of</strong> the elastic seismicmoments <strong>of</strong> interior ones. This was modelled by applying an elastic seismic momentmodification factor equal to 0.5 or 0.25 accordingly.One component <strong>of</strong> seismic action is considered along the X-axis direction.The translational degree <strong>of</strong> freedom (DOF) parallel to the direction <strong>of</strong> the seismic action(UX) is constrained for all nodes on each floor such that <strong>wall</strong>s <strong>and</strong> <strong>frame</strong> share the samedisplacements. Since the building is symmetric with no torsional effects, the translationalhorizontal DOF perpendicular to the direction <strong>of</strong> the seismic action (UZ) <strong>and</strong> therotational DOF in the vertical axis (RY) <strong>and</strong> the horizontal axis parallel to the direction <strong>of</strong>the seismic action (RX) are restrained. The translational DOF in the vertical axis (UY) <strong>and</strong>the rotational DOF in the horizontal axis perpendicular to the direction <strong>of</strong> the seismicaction (RZ) are free.Prismatic beams are used where effective beam width is used for the contribution <strong>of</strong> thestiffness <strong>of</strong> the slab. The effective flange width <strong>of</strong> the T- beams on either side <strong>of</strong> the beamis taken to be 0.6m having a constant width over the whole span <strong>of</strong> the beam. The flangewidth is determined according to Eurocode 2 [CEN 2004a].The strength <strong>and</strong> stiffness <strong>of</strong> the columns or <strong>wall</strong>s are modelled independently in the twobending planes. The axial load variation is taken into account for the variation <strong>of</strong> theflexural properties.Columns support the gravity loads within a tributary area extending up to beam mid-span.All permanent <strong>and</strong> imposed loads per unit floor produce triangular distribution <strong>of</strong> loads onbeams.P-δ effects are considered in the analysis through the linearized geometric stiffness matrix<strong>of</strong> columns.Due to the building’s symmetry only half <strong>of</strong> the building was used in the analysis toreduce computational dem<strong>and</strong>s having a building plan <strong>of</strong> 25m x 12.5m. The beamsperpendicularly connected to the line <strong>of</strong> symmetry have half their length (2.5m) <strong>and</strong> nocolumns are located on the line <strong>of</strong> symmetry. (see Figure 5.4 <strong>and</strong> Figure 5.5)Columns <strong>and</strong> <strong>wall</strong>s are assumed fixed at ground level.Joints are considered rigid.52


Chapter 5: Structural modelling <strong>and</strong> analysis methodsFigure 5.4 Structural model for a five – storey <strong>dual</strong> building taken from ANSRuopFigure 5.5 Structural model for an eight – storey <strong>dual</strong> building taken from ANSRuop5.4. Linear Static Analysis - “Lateral Force Method”The linear elastic (equivalent) static analysis “lateral force method” was performed byPapailia [2011] in order to carry out the design <strong>and</strong> the evaluation <strong>of</strong> the <strong>buildings</strong> for theconstruction <strong>of</strong> the <strong>fragility</strong> curves. The method was performed according to Eurocode 8 –Part 1 [CEN,2004b], where the horizontal component <strong>of</strong> the seismic action is distributed withan assumed linear mode shape along the height <strong>of</strong> the building. This method is applied to<strong>buildings</strong> which are both regular in plan <strong>and</strong> in elevation, if the building response is notaffected by higher modes. The base shear <strong>of</strong> the structure is determined according to the mass<strong>of</strong> the building <strong>and</strong> the design or elastic spectrum at the 1 st translational mode <strong>of</strong> the structure.The design spectrum is used for the design <strong>of</strong> the <strong>buildings</strong> <strong>and</strong> the elastic spectrum for theassessment.WhereV b = m eff S e,d T 1 (5.3)m effis the effective mass <strong>of</strong> the building associated with the gravity loads53


Chapter 5: Structural modelling <strong>and</strong> analysis methodsS e (T 1 ) the elastic horizontal ground acceleration response spectrum at the fundamental periodS d (T 1 ) the design spectrum at the fundamental periodAccording to EC8, the elastic response spectrum S e (T) is defined by:0 ≤ T ≤ T C ∶ S e T = a g S 1 + T. (η 2. 5 − 1) (T B5.4)T B ≤ T ≤ T C ∶ S e T = 2. 5 Sa g η (5.5)T C ≤ T ≤ T D : S e T = 2. 5 Sa g η TcT( 5.6)T D ≤ T ≤ 4s: S e T = 2. 5 Sa g η T CT DT2( 5.7)Where,SηTT ca gis the soil factoris the damping correction factorthe period <strong>of</strong> vibration <strong>of</strong> linear SDOF systemthe corner period <strong>of</strong> the constant spectral acceleration branchthe design ground acceleration on type A groundThe design response spectrum S d (T 1 ) is defined by:0 ≤ T ≤ T C ∶ S d T = S a g 2 3 + TT B. ( 2.5q − 2 3 ) ( 5.8)T B ≤ T ≤ T C ∶ S d T = 2. 5 S a gq( 5.9)T C ≤ T ≤ T D : S d T= 2. 5 S a gqTcT( 5.10)≥ β . a gT D ≤ T ≤ 4s: S d T5.11)= 2. 5 S a gq≥ β . a gT C T DT 2(qβis the behaviour factoris the lower bound factor for the horizontal design spectrum54


Chapter 5: Structural modelling <strong>and</strong> analysis methodsAll spectrums are computed by taking spectrum as Type 1 <strong>of</strong> soil class C, thus T C =0.6sec <strong>and</strong>the soil factor is 1.15.For the computation <strong>of</strong> the fundamental period <strong>of</strong> the structure the Rayleigh quotient is beingused:Where,T 1 = 2π m iδ i2F i δ i(5.12)im iF iδ iis the index <strong>of</strong> the degree <strong>of</strong> freedom,is the mass <strong>of</strong> the floorsis the lateral force applied to the corresponding degree <strong>of</strong> freedomis the displacement obtained from the elastic analysis.The base shear calculated in (5.3) is distributed along the height <strong>of</strong> the building. Thedistribution <strong>of</strong> the lateral forces is given by:F i = V bz i m iz j m j(5.13)Where z i , z j is the height <strong>of</strong> the masses m i , m j above the level <strong>of</strong> application <strong>of</strong> the seismicaction. According to EC8, if T 1


Chapter 5: Structural modelling <strong>and</strong> analysis methodsP-δ effects are considered in the analysis.56


Chapter 6: Assessment <strong>of</strong> Buildings6. ASSESMENT OF BUILDINGSThe assessment <strong>of</strong> the <strong>buildings</strong> is done by the procedure according to Eurocode 8 - Part 3[CEN, 2005]. The estimation <strong>of</strong> the damage measure capacities for each limit state <strong>and</strong>computation <strong>of</strong> the damage measure dem<strong>and</strong>s for each analysis method are described in thischapter. Two limit states are considered as specified by EC8 Part3 [CEN,2005] for “DamageLimitation” which accounts for the yielding <strong>of</strong> the elements <strong>and</strong> the “No collapse” state whichaccounts for the ultimate or collapse limit <strong>of</strong> the elements. The equations given in this chapterare used for the assessment <strong>of</strong> the <strong>buildings</strong> <strong>and</strong> are adopted in the computational s<strong>of</strong>twareANSRuop <strong>and</strong> by Papailia [2011] for the simplified analysis using the lateral force method.6.1. Limit State <strong>of</strong> Damage Limitation (DL)According to Eurocode 8 Part 3 [CEN, 2005], the capacity used for this limit state is theyielding bending moment under the design value <strong>of</strong> the axial force. In order to compute theyielding moment <strong>of</strong> the members, first the yield curvature should be calculated, which isidentified with the yielding <strong>of</strong> the tension reinforcement.The yield curvature, φ y , is given by:φ y =f yLE s 1−ξ y d( 6.1)Where,f yLis the yield stress <strong>of</strong> the longitudinal barsξ y is the neutral axis depth at yielding (normalized to the section effective depth, d),given by:ξ y = (a 2 A 2 + 2aB) 1/2 − aA ( 6.2)Where,aA <strong>and</strong> Bis the ratio <strong>of</strong> the elastic moduli (steel to concrete) , Esare given by:Ec57


Chapter 6: Assessment <strong>of</strong> BuildingsΑ = ρ 1 + ρ 2 + ρ v +Where,N<strong>and</strong> Β = ρbdf 1 + ρ 2 δ 1 + ρ v(1+δ 1 )+ N( 6.3)y 2 bd f yρ 1 <strong>and</strong> ρ 2ρ vδ 1Nthe ratios <strong>of</strong> the tension <strong>and</strong> compression reinforcement respectively. The area<strong>of</strong> any diagonal steel reinforcement is added multiplied by the cosine <strong>of</strong> theirangle.the ratio <strong>of</strong> the web reinforcementthe ratio <strong>of</strong> the distance <strong>of</strong> the centre <strong>of</strong> compression reinforcement from theextreme compression fibre to the width <strong>of</strong> the compression zone, d 1b .is the axial loadFor members <strong>of</strong> high axial load ratio, ν=Ν/Α c f c , the curvature is:φ y =1.8f cE c ξ y d( 6.4)where the neutral axis depth at yielding, ξ y , is the same as before, but A <strong>and</strong> B becomesΑ = ρ 1 + ρ 2 + ρ v −N<strong>and</strong> Β = ρ1.8αbd f 1 + ρ 2 δ 1 + ρ v(1+δ 1 )c 2( 6.5)The lower <strong>of</strong> the two φ y values becomes the yield curvature. Thus the yield moment can becomputed as:2Μ y= φ ξ ybd 3 y {E c21+δ 12− ξ y3+ E s 1−δ 121 − ξ y ρ 1 + ξ y − δ 1 ρ 2 + ρ v61 − δ 1 ( 6.6)The chord rotation at yielding according to Biskinis <strong>and</strong> Fardis [2010], adopted in Eurocode 8– Part 3 is evaluated by:For beams <strong>and</strong> columns with rectangular sections,θ y = φ yL V +a v z3+ 0.0014 1 + 1.5 L V+ a slφ y d b f y8 f c( 6.7)For <strong>wall</strong>s,θ y = φ yL v +a v z3+ 0.0013 + a slφ y d b f y8 f c( 6.8)Where,58


Chapter 6: Assessment <strong>of</strong> Buildingsφ ya v zis the yield curvature <strong>of</strong> the end sectionis the tension drift <strong>of</strong> the bending moment diagram where:o av = 1, if yield moment at the section exceeds the product <strong>of</strong> L V <strong>and</strong> the shearresistance <strong>of</strong> the member considered without shear reinforcement according toEurocode 2 (CEN 2004). M y > V R,c L v . av = 0 if otherwise.o z is the length <strong>of</strong> the internal lever arm taken equal to z = d-d 1 in beams <strong>and</strong>columns, z = 0.8h in <strong>wall</strong>s with rectangular section.a sl a sl =1 if slippage <strong>of</strong> longitudinal bars from anchorage zone beyond theend section is possible. The contribution <strong>of</strong> bar pull-out from joints to the fixedend rotation at member ends is considered when a sl =1.a sl =0if slippage is not possiblef y <strong>and</strong> f cdL s /hd bLsteel yield stress <strong>and</strong> concrete strength respectivelythe effective depth <strong>of</strong> the full section.shear span ratiothe mean diameter <strong>of</strong> the tension reinforcement.The first term <strong>of</strong> the above equations relate to the theoretical yield curvature. It takes intoaccount the shift rule where the yielding <strong>of</strong> the tension reinforcement shifts up to the point <strong>of</strong>the first diagonal crack leading to an increase in yield chord rotation. The second term <strong>of</strong> theabove expression relates to the experimental chord rotation at flexural yielding <strong>and</strong> the thirdterm <strong>of</strong> the expression accounts for the slippage <strong>of</strong> the longitudinal bars from the anchoragezone to the end <strong>of</strong> the section.For verifications carried out in terms <strong>of</strong> deformations, deformation dem<strong>and</strong>s obtained fromthe analysis <strong>of</strong> the structural model require the use <strong>of</strong> the estimation <strong>of</strong> the effective crackedstiffness <strong>of</strong> concrete at yielding. Thus according to EC8 [CEN,2005] the secant stiffness to themember yield-point is used:Where,EI eff = M y L V3θ y( 6.9)M yis the yield moment using the mean material strengths.L V is the member shear span which is the ratio <strong>of</strong> M/V at the member ends, thus it is thedistance <strong>of</strong> the member end to the point <strong>of</strong> zero moments.θ yis the yield chord rotation59


Chapter 6: Assessment <strong>of</strong> Buildings6.2. Limit State <strong>of</strong> Near Collapse (NC)The value <strong>of</strong> the total chord rotation capacity at ultimate <strong>of</strong> concrete members under cyclicloading is taken from Biskinis <strong>and</strong> Fardis [2010] which is also adopted in Eurocode 8- Part 3.The flexure-controlled ultimate chord rotation is equal to:plθ u = θ y + θ um( 6.10)Whereplθ umthe plastic part <strong>of</strong> the chord rotation capacity <strong>of</strong> concrete members under cyclicloadingplθ um=bwa st 1 − 0.525a cy 1 + 0.6a sl 1 −0.052max 1.5; min 10; hbw0.2 v max 0.01;ω 2min(9; Lv ) 1/3 f 0.2 max 0.01;ω 1 h c 25 αρ fywsx fc 1.225 100ρ d( 6.11)Where,bwa sta cya slhL V =M/Vνω 1 , ω 2Where,f c <strong>and</strong> f ywis equal to 0.022 for heat-treated (Tempcore) steelis equal to zero for monotonic loading <strong>and</strong> one for cyclic loading.is equal to one if there is slip in the longitudinal reinforcement bars from theiranchorage beyond the section <strong>of</strong> maximum moment or zero if there is not.is the depth <strong>of</strong> the memberis the shear span ratio at the end <strong>of</strong> the section=N/bhf c where b is the width <strong>of</strong> compression zone <strong>and</strong> N is the axial forceis the mechanical reinforcement ratio <strong>of</strong> the tension <strong>and</strong> compressionlongitudinal reinforcement respectively, including web reinforcementω 1 = ρ 1 + ρ v f yL /f c <strong>and</strong> ω 2 = ρ 2 f yL /f cthe concrete compressive strength <strong>and</strong> the stirrup yield strength (MPa)respectively obtained as mean values.ρ sx =A sx /b w s h is the ratio <strong>of</strong> transverse steel parallel to the direction x <strong>of</strong> the loading,60


Chapter 6: Assessment <strong>of</strong> Buildingss hρ dαis the stirrup spacing.the steel ratio <strong>of</strong> the diagonal reinforcement in each diagonal directionthe confinement effectiveness factor which is equal to:α = 1 − s 2b o1 − s 2 o1 − b i 26 o b o( 6.12)Where, o <strong>and</strong> b ob ithe dimension <strong>of</strong> confined core to the centreline <strong>of</strong> the hoopthe centreline spacing <strong>of</strong> longitudinal bars laterally restrained by a stirrupcorner or a cross tie along the perimeter <strong>of</strong> the cross section.According to Eurocode 8 - Part 3 [CEN, 2005], the cyclic shear strength, V R as controlled bythe stirrups, for beams, columns <strong>and</strong> <strong>wall</strong>s is according to the following expression. (units areMN <strong>and</strong> meters).−xplV R = min N; 0.55A2L c f c + 1 − 0.05min 5; μV∆. 0.15max 0.5; 100ρ tot 1 −0.16min5;LVfcAc+VW ( 6.13)Where,hxL V =M/VNA cf cρ totplμ ∆V wis the depth <strong>of</strong> the cross sectionis the compression zone depthis the ratio <strong>of</strong> moment/shear at the end <strong>of</strong> the sectionis the compression axial forceis the cross sectional area taken as b w d for a rectangular web <strong>of</strong> width b w <strong>and</strong>structural depth <strong>of</strong> d.the concrete compressive strength (MPa) obtained as mean values. For primaryseismic elements it is divided by a partial factor for concrete.the longitudinal reinforcement ratiothe plastic dem<strong>and</strong> <strong>of</strong> ductility dem<strong>and</strong>, which is the ratio <strong>of</strong> the plastic part <strong>of</strong>the chord rotation, ζ, normalized to the chord rotation at yielding, ζ y .is the contribution <strong>of</strong> the transverse reinforcement to shear resistance takenequal to61


Chapter 6: Assessment <strong>of</strong> BuildingsV W = ρ w b w zf yw ( 6.14)Where,ρ wzf ywthe transverse reinforcement ratiolength <strong>of</strong> the internal lever armyield stress <strong>of</strong> the transverse reinforcement. For primary seismic elements it is dividedby the partial factor for steelThe shear strength <strong>of</strong> a concrete <strong>wall</strong>, V R , should not exceed the value which corresponds tothe failure due to web crushing, V R,max . This limit under cyclic loading is given by thefollowing expression (units are in MN <strong>and</strong> meters)V R,max =0.85 1 − 0.06min 5; μ ∆pl1 + 1.8min 0.15; 1 +A c f c0.25max⁡ 1.75;100ρtot)1−0.2min2;LVfcbwzNf c is in MPa, b w <strong>and</strong> z are in meters <strong>and</strong> V R,max is in MN.( 6.15)If web crashing occurs prior to flexural yielding then the shear strength under cyclic loading isobtained when μ ∆ pl = 0.If the shear span ratio at the end section in a concrete column is less than or equal to 2 (L s /h ≤2.0) then its shear strength, V R , should not exceed the value which corresponds to failure bythe crushing <strong>of</strong> the web along the diagonal <strong>of</strong> the column after flexural yielding, V R,max ,which under cyclic loading may be calculated as:V R,max =47 1 − 0.02min 5; μ ∆pl1 + 1.35 NA c f c1 + 0.45⁡(100ρ tot ) min 40; f c b w z sin2δ( 6.16)Where δ is the angle between the diagonal <strong>and</strong> the axis <strong>of</strong> the column: tan δ =2L V.62


Chapter 6: Assessment <strong>of</strong> Buildings6.3. Estimation <strong>of</strong> damage measure dem<strong>and</strong>sDem<strong>and</strong>s are obtained from the analysis <strong>of</strong> the structural model for the seismic actiondepending on the analysis method. It is reminded that the damage measure dem<strong>and</strong>s in thisstudy are the peak chord rotation <strong>and</strong> shear force dem<strong>and</strong>s. The peak chord rotation is definedas the member drift ratio; the deflection at the end <strong>of</strong> the shear span divided by the shear span.In the nonlinear time-history analysis the <strong>wall</strong> shear force dem<strong>and</strong>s in <strong>wall</strong>-<strong>frame</strong> <strong>buildings</strong>are not amplified to capture the effects <strong>of</strong> higher modes since they are taken into account inthe analysis. In the lateral force method by Papailia [2011] once plastic hinge starts forming inthe base <strong>of</strong> the <strong>wall</strong> the shear force dem<strong>and</strong>s are amplified to take into account higher modeeffects according to the proposal in Keintzel [1990] adopted also by CEN [2004a] for DC H<strong>wall</strong>s. Once plastic hinges starts forming in the structure the shear forces in beams <strong>and</strong>columns are calculated from the plastic mechanism <strong>and</strong> the yield moments <strong>of</strong> the sections.63


Chapter 7: Methodology <strong>of</strong> Fragility Analysis7. METHODOLOGY OF FRAGILITY ANALYSISThe seismic <strong>fragility</strong> curves <strong>of</strong> regular reinforced concrete <strong>frame</strong> <strong>and</strong> <strong>wall</strong>-<strong>frame</strong> <strong>buildings</strong> arestudied. Three-dimensional models <strong>of</strong> the full <strong>buildings</strong> are used in order to construct the<strong>fragility</strong> curves using the nonlinear static “pushover” analysis (SPO) <strong>and</strong> the dynamic analysis(IDA). These results are then compared against the <strong>fragility</strong> curves obtained using the “lateralforce method” by Papailia [2011]. The results are presented in terms <strong>of</strong> <strong>fragility</strong> curves fortwo member limit states <strong>of</strong> yielding <strong>and</strong> ultimate deformation in bending or shear.7.1. Damage MeasuresThe damage measures (DM) used in order to obtain the <strong>fragility</strong> curves in this study are thechord rotations <strong>and</strong> the shear force dem<strong>and</strong>s. The chord rotations are found for the twodamage states <strong>of</strong> yielding <strong>and</strong> ultimate conditions <strong>and</strong> the shear forces are found for theultimate condition due to shear failure. The shear forces are taken from outside or inside theplastic hinge. The mean values for the capacities <strong>of</strong> the two damage states are obtained usingEurocode 8 Part 3 [CEN,2005] <strong>and</strong> are consistent with the capacities for flexure <strong>of</strong> Biskinis<strong>and</strong> Fardis [2010a,b] <strong>and</strong> for shear <strong>of</strong> Biskinis et al. [2004] as presented in Chapter 6.The values for DM-dem<strong>and</strong> for each member are obtained through the deterministic seismicanalysis (for the three seismic analysis methods).The damage measure dem<strong>and</strong>s obtained from the LFM are taken for each IM througha deterministic static analysis using an inverted triangular pattern as presented inPapailia [2011].In the dynamic analysis (IDA) the mean damage measure dem<strong>and</strong>s from the 14 semiartificialaccelerogram dynamic analyses are obtained for each IM (i.e. the excitationPGA).The damage measure dem<strong>and</strong>s obtained for the SPO analysis are obtained fromdeterministic nonlinear static analysis using the inverted triangular distribution pattern.All analysis procedures follow the methods <strong>and</strong> approaches provided by CEN (2005) <strong>and</strong>the mean material properties were used (f cm =f ck +8MPa <strong>and</strong> f ym =1.15f yk , see Section 3.3).64


Chapter 7: Methodology <strong>of</strong> Fragility Analysis7.2. Exclusion <strong>of</strong> unrealistic results for IDACertain damage measure dem<strong>and</strong>s obtained from IDA are much higher than the capacities <strong>of</strong>the members. This may lead to erroneous response estimates. This error comes fromnumerical instability thus this may lead to unrealistic response values. These values need to beneglected when calculating the mean <strong>and</strong> variance values <strong>of</strong> these damage indices which arerequired to construct the <strong>fragility</strong> curves. Therefore, damage indices (ratio <strong>of</strong> the DMdem<strong>and</strong>sto DM-capacities) larger than a threshold <strong>of</strong> 200% <strong>of</strong> the mean damage indices perIM (i.e. excitation PGA) are neglected when calculating the statistical parameters. Figure 7.1shows an example where the damage indices above the continuous line on the plot (i.e. thethreshold) are neglected. The zero-value damage indices are due to incomplete analyses <strong>and</strong>are also neglected.Figure 7.1 Exclusion <strong>of</strong> unrealistic results in IDA (damage indices above continuous line are7.3. Determination <strong>of</strong> variabilityneglected)The coefficient <strong>of</strong> variation (CoV) reflects all the variability <strong>and</strong> uncertainty regarding theused models, materials <strong>and</strong> geometries <strong>and</strong> the characteristics <strong>of</strong> seismic input.The variation <strong>of</strong> the DM-capacities reflects the uncertainty in the models that are used toestimate the mean capacity values <strong>and</strong> the scatter <strong>of</strong> the material <strong>and</strong> the geometric properties.These CoV values are taken from Biskinis et al. [2004] <strong>and</strong> Biskinis <strong>and</strong> Fardis [2010a,b], <strong>and</strong>are presented in Table 7.1.The CoV values for the DM-dem<strong>and</strong>s used for the SPO <strong>and</strong> LFM are different than the onestaken for IDA:65


Chapter 7: Methodology <strong>of</strong> Fragility AnalysisThe CoV values for the damage measure dem<strong>and</strong>s used for IDA are found explicitlyfrom the analysis. In the dynamic analysis the variability <strong>of</strong> the DM-dem<strong>and</strong> <strong>of</strong> the 14semi-artificial accelerograms cover the variability <strong>of</strong> the ground motion <strong>and</strong> <strong>of</strong> damagemeasure dem<strong>and</strong>.For the computation <strong>of</strong> the <strong>fragility</strong> curves using the LFM <strong>and</strong> SPO the CoV valuesfor DM-dem<strong>and</strong>s cannot be found explicitly from the analysis. Thus the CoV valuesfor the chord rotation dem<strong>and</strong>s are based on extensive comparisons <strong>of</strong> inelastic to theirelastic estimates <strong>of</strong> chord rotation dem<strong>and</strong>s in height wise regular multi-storey<strong>buildings</strong> by Panayiotakos <strong>and</strong> Fardis [1999], Kosmopoulos <strong>and</strong> Fardis [2007]. Thecoefficient <strong>of</strong> variation values for the shear force dem<strong>and</strong>s listed are based onparametric studies. These CoV values are presented in Table 7.2.The CoV values per storey in terms <strong>of</strong> intensity measure (i.e. PGA) obtained from IDA arepresented in Appendix C1. On the same plots the straight line represents the CoV values forthe damage measure dem<strong>and</strong>s <strong>and</strong> the CoV values <strong>of</strong> the spectral value taken from Table 7.2.Figure 7.2 <strong>and</strong> Figure 7.3 illustrate examples <strong>of</strong> the dispersion values per IM (i.e. PGA) for a<strong>frame</strong> <strong>and</strong> a <strong>wall</strong>-<strong>frame</strong> <strong>dual</strong> building. It can be observed that the CoV-values determinedthrough IDA are lower than the ones determined from previous studies (shown in a straightline on the plot representing CoV-values <strong>of</strong> DM-dem<strong>and</strong> <strong>and</strong> spectral value). Also the me<strong>and</strong>ispersions <strong>of</strong> DM-dem<strong>and</strong>s for beams <strong>and</strong> columns are slightly higher in <strong>wall</strong>-<strong>frame</strong><strong>buildings</strong> than in <strong>frame</strong>s. There is a larger scatter <strong>of</strong> CoV-values in the storeys <strong>of</strong> <strong>dual</strong><strong>buildings</strong> compared to pure <strong>frame</strong>.66


Chapter 7: Methodology <strong>of</strong> Fragility AnalysisFigure 7.2 Coefficient <strong>of</strong> variation (CoV) <strong>of</strong> DM-dem<strong>and</strong>s for five-storey <strong>frame</strong> building <strong>designed</strong> to DC M<strong>and</strong> PGA=0.20g67


Chapter 7: Methodology <strong>of</strong> Fragility AnalysisFigure 7.3 Coefficient <strong>of</strong> variation (CoV) <strong>of</strong> DM-dem<strong>and</strong>s for five-storey <strong>frame</strong>-equivalent building<strong>designed</strong> to DC M <strong>and</strong> PGA=0.20g68


Chapter 7: Methodology <strong>of</strong> Fragility Analysis7.4. Construction <strong>of</strong> <strong>fragility</strong> curvesFor the construction <strong>of</strong> <strong>fragility</strong> curves the probability <strong>of</strong> a damage measure (DM) dem<strong>and</strong> toexceed a certain DM-capacity is expressed in terms <strong>of</strong> Peak Ground Acceleration (PGA).PGA was used instead <strong>of</strong> other intensity measures in order to be consistent with the use <strong>of</strong> thedesign acceleration as a design parameter.As mentioned previously the members’ fragilities are expressed for the damage states <strong>of</strong>yielding <strong>and</strong> ultimate. The member yielding or ultimate damage state in flexure is reachedwhen the chord rotation at the member end exceeds the yielding or ultimate flexural capacity.The shear failure is when the shear force exceeds the shear capacity <strong>of</strong> the member, where theshear capacity is a function <strong>of</strong> the rotation ductility dem<strong>and</strong> at the member end.The <strong>fragility</strong> <strong>of</strong> the member is obtained for each IM (i.e. PGA) from deterministic analysis<strong>and</strong> is the conditional-on-IM probability that the dem<strong>and</strong> <strong>of</strong> the given damage measure willexceed its capacity. It is assumed that the <strong>fragility</strong> curves are expressed in log-normaldistribution. Based on this assumption the cumulative probability <strong>of</strong> occurrence can beexpressed as:P D′ ≥ C = 1 − Φln λβ 2 D +β 2PGA +β 2C( 7.1)Where,D’ is the damage measure dem<strong>and</strong> (DM- dem<strong>and</strong>)CλΦis the threshold damage measure capacity for a limit state (DM- capacity)is the mean damage index for each IM. The damage indices obtained usingIDA are the mean <strong>of</strong> the 14 damage indices per IM (i.e. PGA). The damageindices for the SPO <strong>and</strong> LFM are found from the analysis.is the st<strong>and</strong>ard normal distributionβ C <strong>and</strong> β D are the st<strong>and</strong>ard deviation for the capacity <strong>and</strong> the dem<strong>and</strong>, such that β C =ln⁡(1 + δ C2) <strong>and</strong> β D = ln⁡(1 + δ D2).β PGA it is the st<strong>and</strong>ard deviation for the spectral value (δ PGA ) given in Table 7.2.2β PGA = ln⁡(1 + δ PGA ). It is not used for IDA, since dispersion is takenexplicitly from the analysis; i.e. β PGA = 0 for IDA.δ C is the coefficient <strong>of</strong> variation for the DM-capacity values found in Table 7.1.δ Dis the coefficient <strong>of</strong> variation for the DM-dem<strong>and</strong> values <strong>and</strong> values fromTable 7.2. For the IDA method they are found explicitly from the analysis.69


Chapter 7: Methodology <strong>of</strong> Fragility Analysisδ Dis the coefficient <strong>of</strong> variation for the spectral value.Table 7.1 Values <strong>of</strong> coefficient <strong>of</strong> variation for DM-capacity valuesCapacityCoVδ C1 Beam or column chord rotation at yielding 0.33δ C2 Beam or column chord rotation at ultimate 0.38δ C3Shear resistance in diagonal tension (inside or outsideplastic hinge)0.15δ C4 Wall chord rotation at yielding <strong>of</strong> the base 0.40δ C5 Wall chord rotation at ultimate <strong>of</strong> the base 0.32δ C6 Wall shear resistance in diagonal compression 0.175Table 7.2 Values <strong>of</strong> coefficient <strong>of</strong> variation for DM-dem<strong>and</strong> valuesδ D1δ D2δ D3δ D4δ D5δ D6δ PGADem<strong>and</strong>Beam chord rotation dem<strong>and</strong>, for given spectralvalue at the fundamental periodColumn chord rotation dem<strong>and</strong>, for given spectralvalue at the fundamental periodWall chord rotation dem<strong>and</strong>, for given spectralvalue at the fundamental periodBeam shear force dem<strong>and</strong>, for given spectral valueat the fundamental periodColumn shear force dem<strong>and</strong>, for given spectralvalue at the fundamental periodWall shear force dem<strong>and</strong>, for given spectral valueat the fundamental periodSpectral value, for given PGA <strong>and</strong> fundamentalperiodCoV0.250.200.250.100.150.200.2570


Chapter 8: Results <strong>and</strong> Discussion8. RESULTS AND DISCUSSIONThe results from the analysis <strong>of</strong> the structural models <strong>and</strong> the member <strong>fragility</strong> curves for thetypes <strong>of</strong> <strong>buildings</strong> examined are discussed in this chapter. Section 8.1 presents the modalanalysis results from the three-dimensional structural models. Section 8.2 indicates themedian PGAs (g) at attainment <strong>of</strong> the damage states <strong>of</strong> each member for the three analysismethods. Section 8.3 discusses the <strong>fragility</strong> results for <strong>wall</strong>-<strong>frame</strong> <strong>dual</strong> systems <strong>and</strong> Section8.4 for <strong>frame</strong> systems. The differences in <strong>fragility</strong> curves according to the different designparameters (see Section 3.1) are further discussed. Section 8.5 presents the comparison <strong>of</strong> themember <strong>fragility</strong> curves for the three different analysis methods. It is reminded that themethods <strong>of</strong> analysis include the Incremental Dynamic Analysis (IDA) <strong>and</strong> the Static PushoverAnalysis (SPO) <strong>and</strong> these were compared against a simplified method using the Lateral forcemethod (LFM) by Papailia [2011].In the current chapter only indicative results will be shown in order to draw conclusions onthe results <strong>of</strong> the analysis. Appendix A1 presents the member <strong>fragility</strong> curves <strong>of</strong> all theexamined <strong>buildings</strong> analysed using IDA <strong>and</strong> in Appendix A2 using the SPO analysis.Appendix A3 presents the <strong>wall</strong> member <strong>fragility</strong> curves for shear ultimate state for thedifferent methods. In LFM analysis <strong>wall</strong> <strong>fragility</strong> curves for shear failure include results with<strong>and</strong> without inelastic amplifications to take into account higher mode effects.Appendix B1 presents the member <strong>fragility</strong> curves for the three different methods <strong>and</strong>Appendix B2 presents the comparison <strong>of</strong> the member <strong>fragility</strong> curves for the most criticalmember for the three methods. Appendix C1 presents the coefficient <strong>of</strong> variation values perIM (i.e. excitation PGA) used in the construction <strong>of</strong> the <strong>fragility</strong> curves <strong>and</strong> Appendix C2presents the damage indices (ratio <strong>of</strong> DM-dem<strong>and</strong>s to DM-capacities) per IM for eachmember.71


Chapter 8: Results <strong>and</strong> Discussion8.1. Modal analysis resultsThe modal periods <strong>and</strong> participating masses <strong>of</strong> the structural model used for the three modeswith the largest modal participation mass percentage is given in the following tables. Themodal periods <strong>of</strong> the structure are obtained using the effective stiffness <strong>of</strong> the members usingthe structural s<strong>of</strong>tware <strong>of</strong> ANSRuop.Table 8.1 Modal periods <strong>and</strong> participating masses for <strong>frame</strong> systemsStoreysDesignPGA5 0.20g M 1235 0.25g M 1235 0.25g H 123DC Mode T (sec) Effective modalmass (%)1.910.560.271.720.520.261.690.510.2678.1712.725.3179.2111.885.3580.0811.464.98Table 8.2 Modal periods <strong>and</strong> participating masses for <strong>frame</strong>-equivalent <strong>dual</strong> systemsStoreysDesignPGA5 0.20g M 1235 0.25g M 1235 0.25g H 1238 0.20g M 1238 0.25g M 123DC Mode T (sec) Effective modalmass (%)1.990.560.261.660.450.201.630.460.222.610.720.332.500.700.3475.4213.226.0373.6014.375.6274.413.446.3270.8713.926.2271.7213.035.9872


Chapter 8: Results <strong>and</strong> DiscussionTable 8.3 Modal periods <strong>and</strong> participating masses for <strong>wall</strong>-equivalent <strong>dual</strong> systemsStoreysDesignPGA5 0.20g M 1235 0.25g M 1235 0.25g H 1238 0.20g M 1238 0.25g M 123DC Mode T (sec) Effective modalmass (%)1.830.480.231.460.370.151.500.400.172.490.670.292.320.670.3373.1214.596.6371.5215.846.6472.3915.196.6469.6214.586.5069.1313.686.45Table 8.4 Modal periods <strong>and</strong> participating masses for <strong>wall</strong> <strong>dual</strong> systemsStoreysDesignPGA5 0.20g M 1235 0.25g M 1235 0.25g H 1238 0.20g M 1238 0.25g M 1238 0.25g H 123DC Mode T (sec) Effective modalmass (%)1.620.390.161.240.280.111.150.250.102.110.510.211.920.460.191.570.330.1370.6616.466.8869.2217.657.1968.5618.147.3067.5716.126.7767.3416.456.8165.2217.777.1773


Chapter 8: Results <strong>and</strong> DiscussionThe modal analysis results illustrate that the <strong>buildings</strong> <strong>designed</strong> for higher design peak groundacceleration or ductility class generally slightly reduces the fundamental period <strong>of</strong> thestructure is; i.e. making the structure stiffer. As the proportion <strong>of</strong> total base shear taken by the<strong>wall</strong>s increases, the effective modal mass percentage decreases at the first mode <strong>and</strong> increasesfor higher modes. Design for a higher PGA reduces effective modal mass percentage at thefundamental period <strong>and</strong> increases at the higher modes.8.2. Median PGAs at attainment <strong>of</strong> the damage state for the three methodsThe median PGAs at attainment <strong>of</strong> the damage states for the members indicate clearly thedifferences between the three analysis methods <strong>and</strong> the differences when designing todifferent design parameters (Table 8.5 to Table 8.11). The median PGAs indicate the PGAvalues for 50% probability <strong>of</strong> exceeding a certain damage state in each member. A dash (-)indicates that the median PGA is larger than 1g. Member median PGA at attainment <strong>of</strong> thedamage state is presented for members in flexure <strong>and</strong> in shear. Discussion on the resultsshown in these tables is made in Section 8.3, Section 8.4 <strong>and</strong> Section 8.5.designPGA74designPGATable 8.5 Median PGA (g) at attainment <strong>of</strong> the damage state in 5-storey <strong>frame</strong> systemsDCAnalysismethodBeamYieldingBeamUltimate(flex)BeamUltimate(shear)ColumnYieldingColumnUltimate(flex)ColumnUltimate(shear)0.20g M LFM 0.14g 0.65g - 0.84g - -SPO 0.12g 0.70g - 0.69g - -IDA 0.14g 0.74g - 0.85g - -0.25g M LFM 0.16g 0.79g - 0.74g - 0.95gSPO 0.16g 0.78g - 0.70g - -IDA 0.19g 0.74g - 0.81g - -0.25g H LFM 0.13g 0.70g - 0.68g - -SPO 0.17g 0.80g - 0.60g - -IDA 0.19g 0.83g - 0.64g - -Table 8.6 Median PGA (g) at attainment <strong>of</strong> the damage state in 5-storey <strong>frame</strong>-equivalent systemsDCAnalysismethodBeamYieldingBeamUltimate(flex)ColumnYieldingColumnUltimate(flex)WallYieldingWallUltimate(flex)WallUltimate(shear)0.20g M LFM 0.14g 0.62g 0.35g - 0.09g 0.35g 0.25gSPO 0.13g 0.75g 0.42g 0.82g 0.06g 0.29g -IDA 0.18g 0.82g 0.52g - 0.08g 0.38g 0.94g0.25g M LFM 0.18g 0.83g 0.39g 0.94g 0.11g 0.43g 0.19gSPO 0.19g 0.68g 0.52g - 0.09g 0.39g -IDA 0.22g 0.75g 0.46g - 0.10g 0.43g 0.39g0.25g H LFM 0.14g 0.83g 0.38g - 0.10g 0.44g 0.38gSPO 0.16g 0.77g 0.44g - 0.09g 0.41g -IDA 0.19g 0.94g 0.44g - 0.10g 0.47g 0.90g


Chapter 8: Results <strong>and</strong> DiscussiondesignPGATable 8.7 Median PGA (g) at attainment <strong>of</strong> the damage state in 5-storey <strong>wall</strong>-equivalent <strong>dual</strong> systemsDCAnalysismethodBeamYieldingBeamUltimate(flex)ColumnYieldingColumnUltimate(flex)WallYieldingWallUltimate(flex)WallUltimate(shear)0.20g M LFM 0.14g 0.73g 0.38g 0.98g 0.10g 0.38g 0.19gSPO 0.18g 0.87g 0.46g 0.95g 0.09g 0.37g -IDA 0.19g 0.71g 0.45g - 0.11g 0.45g 0.42g0.25g M LFM 0.19g 0.95 0.43g - 0.11g 0.44g 0.18gSPO 0.25g - 0.53g - 0.11g 0.43g -IDA 0.25g 0.84g 0.46g - 0.13g 0.46g 0.37g0.25g H LFM 0.14g 0.83g 0.38g - 0.10g 0.44g 0.38gSPO 0.20g 0.81g 0.48g - 0.09g 0.45g -IDA 0.22g 0.76g 0.40g - 0.11g 0.45g 0.53gdesignPGATable 8.8 Median PGA (g) at attainment <strong>of</strong> the damage state in 5-storey <strong>wall</strong> systemsDCAnalysismethodBeamYieldingBeamUltimate(flex)ColumnYieldingColumnUltimate(flex)WallYieldingWallUltimate(flex)WallUltimate(shear)0.20g M LFM 0.17g 0.84g 0.42g - 0.11g 0.41g 0.17gSPO 0.19g 0.78g 0.47g - 0.11g 0.39g -IDA 0.20g 0.65g 0.38g - 0.13g 0.39g 0.33g0.25g M LFM 0.22g - 0.47g - 0.13g 0.53g 0.19gSPO 0.29g - 0.61g - 0.16g 0.54g -IDA 0.24g - 0.51g - 0.18g 0.39g 0.29g0.25g H LFM 0.25g - 0.57g - 0.17g 0.71g 0.51gSPO 0.24g 0.95g 0.72g - 0.18g 0.62g -IDA 0.28g - 0.54g - 0.20g 0.57g 0.72gTable 8.9 Median PGA (g) at attainment <strong>of</strong> the damage state in 8-storey <strong>frame</strong>-equivalent <strong>dual</strong> systemsdesignPGADCAnalysismethodBeamYieldingBeamUltimate(flex)ColumnYieldingColumnUltimate(flex)WallYieldingWallUltimate(flex)WallUltimate(shear)0.20g M LFM 0.14g 0.64g 0.35g - 0.09g 0. 31g 0.58gSPO 0.25g 0.86g 0.76g - 0.07g 0.29g -IDA 0.27g 0.84g 0.77g - 0.09g 0.39g 0.88g0.25g M LFM 0.14g 0.78g 0.33g - 0.09g 0.33g 0.51gSPO 0.28g 0.77g 0.71g - 0.06g 0.30g -IDA 0.31g 0.94g 0.77g - 0.09g 0.40g 0.77g75


Chapter 8: Results <strong>and</strong> DiscussiondesignPGATable 8.10 Median PGA (g) at attainment <strong>of</strong> the damage state in 8-storey <strong>wall</strong>-equivalent <strong>dual</strong> systemsDCAnalysismethodBeamYieldingBeamUltimate(flex)ColumnYieldingColumnUltimate(flex)WallYieldingWallUltimate(flex)WallUltimate(shear)0.20g M LFM 0.13g 0.70g 0.44g - 0.11g 0.43g 0.18gSPO 0.24g 0.88g 0.83g - 0.10g 0.47g -IDA 0.29g 0.70g 0.67g - 0.12g 0.53g 0.61g0.25g M LFM 0.20g 0.92g 0.38g - 0.12g 0.48g 0.18gSPO 0.27g 0.82g 0.83g - 0.12g 0.50g -IDA 0.30g 0.81g 0.57g - 0.14g 0.54g 0.67gdesignPGATable 8.11 Median PGA (g) at attainment <strong>of</strong> the damage state in 8-storey <strong>wall</strong> systemsDCAnalysismethodBeamYieldingBeamUltimate(flex)ColumnYieldingColumnUltimate(flex)WallYieldingWallUltimate(flex)WallUltimate(shear)0.20g M LFM 0.17g 0.89g 0.49g - 0.12g 0.47g 0.18gSPO 0.22g 0.84g 0.84g - 0.13g 0.47g -IDA 0.24g 0.74g 0.64g - 0.14g 0.39g 0.29g0.25g M LFM 0.20g - 0.44g - 0.14g 0.53g 0.19gSPO 0.37g 0.94g 0.56g - 0.13g 0.50g -IDA 0.35g 0.84g 0.45g - 0.14g 0.40g 0.35g8.3. Fragility curve results for <strong>wall</strong>-<strong>frame</strong> <strong>dual</strong> systemsThe <strong>fragility</strong> curves <strong>of</strong> members for prototype plan- <strong>and</strong> height-wise very regular reinforcedconcrete <strong>wall</strong>-<strong>frame</strong> <strong>buildings</strong> are discussed in this section for the results obtained from IDA<strong>and</strong> SPO. Parameters that were studied include the number <strong>of</strong> storeys, the level <strong>of</strong> Eurocode 8design (in terms <strong>of</strong> design peak ground acceleration <strong>and</strong> ductility class) <strong>and</strong> the percentage <strong>of</strong>seismic base shear taken by the <strong>wall</strong>s. The member <strong>fragility</strong> curves <strong>of</strong> all the <strong>buildings</strong>examined for the analysis performed using nonlinear dynamic analysis are presented inAppendix A1 <strong>and</strong> for nonlinear static pushover analysis in Appendix A2.Figure 8.1, Figure 8.13, Figure 8.14, Figure 8.16, Figure 8.20, Figure 8.21 <strong>and</strong> Figure 8.22refer to examples <strong>of</strong> <strong>wall</strong>-<strong>frame</strong> <strong>buildings</strong>. The first column in each set concerns the beams,the second the column <strong>and</strong> the third the <strong>wall</strong>s. The first row in each set is for yielding <strong>and</strong> thesecond for ultimate state. The <strong>fragility</strong> curves <strong>of</strong> beams <strong>and</strong> columns for the <strong>wall</strong>-<strong>frame</strong>systems are presented for the ultimate state in flexure since it is more critical than shearfailure. In <strong>frame</strong> systems the envelope <strong>of</strong> flexural <strong>and</strong> shear ultimate damage state for beams<strong>and</strong> columns is presented. Fragility curves <strong>of</strong> <strong>wall</strong>s in the ultimate state are the envelope <strong>of</strong> theultimate damage state in flexure <strong>and</strong> shear.76


Chapter 8: Results <strong>and</strong> Discussion1The conclusions for the <strong>wall</strong>-<strong>frame</strong> (<strong>dual</strong>) systems are:0.50 Beams are much 0 more likely 0.2 to reach 0.4 the ultimate 0.6 damage 0.8 state than 1 columns. (seeFigure 8.1).1st 2nd 3rd 4th 5thFigure 8.1 Fragility curves for five-storey <strong>wall</strong>-equivalent building <strong>designed</strong> to PGA=0.20g <strong>and</strong> DC Manalyzed using IDA methodWalls are the most critical members in every design scenario for both yielding <strong>and</strong>ultimate damage state.For the analysis performed using IDA, <strong>wall</strong> failure is usually more critical in shearthan in flexure, except in the following cases:o Eight-storey <strong>frame</strong>-equivalent <strong>and</strong> <strong>wall</strong>-equivalent <strong>dual</strong> systems (see Figure8.2 <strong>and</strong> Figure 8.3).o Five-storey <strong>frame</strong>-equivalent system <strong>designed</strong> to 0.20g <strong>and</strong> DC M (see Figure8.4 (left)).o Five-storey <strong>buildings</strong> <strong>designed</strong> to 0.25g <strong>and</strong> DC H. (see Figure 8.4 (right) <strong>and</strong>Figure 8.5).The <strong>wall</strong> <strong>fragility</strong> curves <strong>of</strong> <strong>wall</strong>-equivalent <strong>dual</strong> <strong>buildings</strong> in the ultimate state aresimilar for shear <strong>and</strong> flexure failure. (see Figure 8.2 to Figure 8.4)77


Chapter 8: Results <strong>and</strong> DiscussionFigure 8.2 Fragility curves <strong>of</strong> <strong>wall</strong>s for eight-storey <strong>frame</strong>-equivalent (left) <strong>and</strong> <strong>wall</strong>equivalentbuilding (right) <strong>designed</strong> to PGA=0.20g <strong>and</strong> DC M analyzed using IDA methodFigure 8.3 Fragility curves <strong>of</strong> <strong>wall</strong>s for eight-storey <strong>frame</strong>-equivalent (left) <strong>and</strong> <strong>wall</strong>equivalentbuilding (right) <strong>designed</strong> to PGA=0.25g <strong>and</strong> DC M analyzed using IDA methodFigure 8.4 Fragility curves <strong>of</strong> <strong>wall</strong>s for five-storey <strong>frame</strong>-equivalent building <strong>designed</strong> to PGA=0.20g <strong>and</strong>DC M (left) <strong>and</strong> <strong>wall</strong> building <strong>designed</strong> to DC H <strong>and</strong> PGA=0.25g (right) analyzed using IDA method78


Chapter 8: Results <strong>and</strong> DiscussionFigure 8.5 Fragility curves <strong>of</strong> <strong>wall</strong>s for five-storey <strong>frame</strong>-equivalent (left) <strong>and</strong> <strong>wall</strong>-equivalent(right) <strong>buildings</strong> <strong>designed</strong> to DC H <strong>and</strong> PGA=0.25g analyzed using IDA methodThe <strong>fragility</strong> curves <strong>of</strong> beams <strong>and</strong> columns, for the results taken from IDA method, arepresented for <strong>frame</strong>-equivalent, <strong>wall</strong>-equivalent <strong>and</strong> <strong>wall</strong> systems in Figure 8.6, Figure8.7, Figure 8.8 <strong>and</strong> Figure 8.9. As it can be observed, the <strong>fragility</strong> results <strong>of</strong> beams forboth damage states show that the middle-storey beams have the highest <strong>fragility</strong>. Thetop-storey beams have the lowest in the yielding state <strong>and</strong> the first-storey beams thelowest in the ultimate state (see Figure 8.6 <strong>and</strong> Figure 8.8). The first-storey columnsare the most critical in five-storey <strong>buildings</strong> (see Figure 8.7) <strong>and</strong> the first- <strong>and</strong> topstoreycolumns in eight-storey <strong>buildings</strong> (see Figure 8.9). The middle-storey columnsare the least fragile in all <strong>buildings</strong> for both damage states (see Figure 8.7 <strong>and</strong> Figure8.9).As the proportion <strong>of</strong> the total base shear taken by the <strong>wall</strong> increases, the <strong>fragility</strong> <strong>of</strong> themiddle-storey columns in both damage states increases. This observation holds forfive- <strong>and</strong> eight-storey <strong>wall</strong>-<strong>frame</strong> <strong>buildings</strong> (see Figure 8.7 <strong>and</strong> Figure 8.9).As the proportion <strong>of</strong> the total base shear taken by the <strong>wall</strong> increases, the <strong>fragility</strong> <strong>of</strong> thelower- <strong>and</strong> top-storey beams does not significantly change in the yielding state <strong>and</strong>increases in the ultimate state. This observation holds for five- <strong>and</strong> eight-storey <strong>wall</strong><strong>frame</strong><strong>buildings</strong> (see Figure 8.6 <strong>and</strong> Figure 8.8).79


Chapter 8: Results <strong>and</strong> Discussion68 x 10-3 1st 2nd 3rd 4th 5th4200 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1Figure 8.6 Beam <strong>fragility</strong> curves for a) yielding <strong>and</strong> b) ultimate state <strong>of</strong> a five-storey <strong>frame</strong>-equivalent(left), <strong>wall</strong>-equivalent 8 x 10-3 (middle) <strong>and</strong> <strong>wall</strong> system (right) building <strong>designed</strong> to DC M <strong>and</strong> PGA=0.20ganalyzed with IDA61st 2nd 3rd 4th 5th4200 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1Figure 8.7 Column <strong>fragility</strong> curves for c) yielding <strong>and</strong> d) ultimate state <strong>of</strong> a five-storey <strong>frame</strong>equivalent(left), <strong>wall</strong>-equivalent (middle) <strong>and</strong> <strong>wall</strong> system (right) building <strong>designed</strong> to DC M <strong>and</strong>PGA=0.20g analyzed with IDA80


Chapter 8: Results <strong>and</strong> Discussion0.40.30.21st 2nd 3rd 4th5th 6th 7th 8th0.100 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1Figure 8.8 Beam <strong>fragility</strong> curves for a) yielding <strong>and</strong> b) ultimate state <strong>of</strong> a eight-storey <strong>frame</strong>-equivalent(left), <strong>wall</strong>-equivalent (middle) <strong>and</strong> <strong>wall</strong> system (right) building <strong>designed</strong> to DC M <strong>and</strong> PGA=0.25g0.40.30.20.1analyzed with IDA1st 2nd 3rd 4th5th 6th 7th 8th00 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1Figure 8.9 Column <strong>fragility</strong> curves for c) yielding <strong>and</strong> d) ultimate state <strong>of</strong> a eight-storey <strong>frame</strong>-equivalent(left), <strong>wall</strong>-equivalent (middle) <strong>and</strong> <strong>wall</strong> system (right) building <strong>designed</strong> to DC M <strong>and</strong> PGA=0.25ganalyzed with IDA81


Chapter 8: Results <strong>and</strong> DiscussionThe differences between <strong>fragility</strong> curves for different design parameters are:1) Design ductility class Design to DC M in lieu <strong>of</strong> DC H the <strong>fragility</strong> <strong>of</strong> beams may reduce againstyielding <strong>and</strong> increase against ultimate state. However, such effects are neithersystematic nor marked (see Figure 8.10, Figure 8.11 <strong>and</strong> Figure 8.12). Design to DC M in lieu <strong>of</strong> DC H column <strong>fragility</strong> is reduced in <strong>frame</strong>-equivalent<strong>and</strong> <strong>wall</strong>-equivalent systems (see Figure 8.10 <strong>and</strong> Figure 8.11) but increased in<strong>wall</strong> systems (see Figure 8.10, Figure 8.11 <strong>and</strong> Figure 8.12). Wall <strong>fragility</strong> in the yielding damage state does not significantly change <strong>and</strong> in theultimate damage state (the envelope <strong>of</strong> flexure <strong>and</strong> shear collapse) is higher for DCM <strong>wall</strong>s. (see Figure 8.10, Figure 8.11 <strong>and</strong> Figure 8.12)82


Chapter 8: Results <strong>and</strong> Discussion0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1DC M DC HFigure 8.10 Fragility curves for most critical members <strong>of</strong> five–storey <strong>frame</strong>-equivalentbuilding <strong>designed</strong> to PGA=0.25g <strong>and</strong> DC M analyzed using IDA method83


Chapter 8: Results <strong>and</strong> Discussion0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1DC M DC HFigure 8.11 Fragility curves for most critical members <strong>of</strong> five–storey <strong>wall</strong>-equivalent building<strong>designed</strong> to PGA=0.25g <strong>and</strong> DC M analyzed using IDA method84


Chapter 8: Results <strong>and</strong> Discussion0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1DC M DC HFigure 8.12 Fragility curves for most critical members <strong>of</strong> five–storey <strong>wall</strong> building <strong>designed</strong>to PGA=0.25g <strong>and</strong> DC M analyzed using IDA method2) Height <strong>of</strong> the building Taller <strong>buildings</strong> exhibit lower fragilities for beams <strong>and</strong> columns <strong>and</strong> similarfragilities for <strong>wall</strong>s in both damage states (see Figure 8.13). The latter is observedexcept in the case <strong>of</strong> the eight-storey <strong>wall</strong> building <strong>designed</strong> to PGA=0.25g <strong>and</strong>DC M which has lower <strong>fragility</strong> for beams but higher <strong>fragility</strong> for columns <strong>and</strong><strong>wall</strong>s. (see Figure 8.14)85


10.5Chapter 8: Results <strong>and</strong> Discussion00 0.2 0.4 0.6 0.8 11st 2nd 3rd 4th 5th10.51st 2nd 3rd 4th00 5th 0.2 6th 0.4 0.6 7th 0.8 8th1Figure 8.13 Member <strong>fragility</strong> curves <strong>of</strong> <strong>frame</strong>-equivalent <strong>dual</strong> systems <strong>designed</strong> to PGA=0.25g <strong>and</strong> DC Mfor: (top) five – storey; (bottom) eight-storey using IDA method86


10.50Chapter 8: Results <strong>and</strong> Discussion 0 0.2 0.4 0.6 0.8 11st 2nd 3rd 4th 5th10.51st 2nd 3rd 4th00 5th 0.2 6th 0.4 0.6 7th 0.8 8th187Figure 8.14 Member <strong>fragility</strong> curves for <strong>wall</strong> systems <strong>designed</strong> to PGA=0.25g <strong>and</strong> DC M curves <strong>of</strong>: (top)five – storey; (bottom) eight-storey using IDA method3) Proportion <strong>of</strong> total base shear taken by the <strong>wall</strong> Wall-equivalent <strong>dual</strong> <strong>and</strong> <strong>wall</strong> <strong>buildings</strong> have similar fragilities for beams <strong>and</strong>columns but lower than <strong>frame</strong>-equivalent <strong>dual</strong> systems since the deformationdem<strong>and</strong> <strong>of</strong> the <strong>frame</strong> is higher. Wall-equivalent <strong>dual</strong> systems are in-between butcloser to <strong>wall</strong> <strong>dual</strong> systems. (see Figure 8.15 - Wall ultimate damage state ispresented for both (f) flexure <strong>and</strong> (g) shear collapse ). As the proportion <strong>of</strong> total base shear taken by the <strong>wall</strong> increases, the <strong>wall</strong>s inyielding <strong>and</strong> ultimate damage state in flexure have lower fragilities but higher forthe ultimate state in shear. (see Figure 8.15).


Chapter 8: Results <strong>and</strong> Discussion0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1FE WE WSFigure 8.15 Member <strong>fragility</strong> curves for a five-storey <strong>frame</strong>-equivalent (FE), <strong>wall</strong>-equivalent (WE), <strong>wall</strong><strong>dual</strong> (WS) system <strong>designed</strong> to PGA=0.20g <strong>and</strong> DC M using SPO method for most critical storey members.88


Chapter 8: Results <strong>and</strong> Discussion4) Design peak ground acceleration (PGA) Design for a higher PGA reduces <strong>fragility</strong> <strong>of</strong> beams in both damage states <strong>and</strong> mayslightly increase <strong>fragility</strong> <strong>of</strong> columns; however this effect is neither systematic normarked. Wall <strong>fragility</strong> is not significantly changed when designing for higherPGA. (see Figure 8.16 <strong>and</strong> Figure 8.17)10.51st 2nd 3rd 4th00 5th 0.2 6th 0.4 0.6 7th 0.8 8th1Figure 8.16 Fragility curves <strong>of</strong> eight–storey <strong>frame</strong>-equivalent building <strong>designed</strong> to DC M <strong>and</strong>: (top)PGA=0.20g; (bottom) PGA=0.25g analyzed using IDA method89


Chapter 8: Results <strong>and</strong> Discussion0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1PGA=0.20g PGA=0.25gFigure 8.17 Member <strong>fragility</strong> curves for a eight-storey <strong>wall</strong>-equivalent system <strong>designed</strong> to DC M <strong>and</strong> forPGA=0.20g <strong>and</strong> PGA=0.25g using IDA method for most critical storey members.90


Chapter 8: Results <strong>and</strong> Discussion8.4. Fragility curve results for <strong>frame</strong> systemsThe construction <strong>of</strong> <strong>fragility</strong> curves <strong>of</strong> plan- <strong>and</strong> height-wise very regular reinforced concrete<strong>frame</strong> <strong>buildings</strong> were also examined. Parameters that were studied include the level <strong>of</strong>Eurocode 8 design (in terms <strong>of</strong> ductility class <strong>and</strong> design PGA). The following conclusionscan be drawn based on the results <strong>of</strong> the <strong>frame</strong> systems:Frames give satisfactory <strong>fragility</strong> results even beyond their design PGAs. (see Table8.5)Beams yield before their design PGA whereas the columns remain elastic well beyondthe design PGA. Also beams are much more likely to reach the ultimate damage statethan columns. (see Figure 8.18 to Figure 8.22 <strong>and</strong> Table 8.5).Design for higher PGA reduces only slightly the fragilities <strong>of</strong> beams <strong>and</strong> columns inyielding damage state <strong>and</strong> may increase <strong>fragility</strong> in the ultimate damage state (seeFigure 8.18).Design to DC M instead <strong>of</strong> DC H may reduce slightly the <strong>fragility</strong> <strong>of</strong> beams <strong>and</strong>columns against yielding, but may increase that <strong>of</strong> beams against ultimate. (see Figure8.19)Wall-<strong>frame</strong> (<strong>dual</strong>) systems have, in general, higher <strong>fragility</strong> than <strong>frame</strong> systems forcolumns <strong>and</strong> lower for beams (see Figure 8.20, Figure 8.21 <strong>and</strong> Figure 8.22).0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1PGA=0.20g PGA=0.25gFigure 8.18 Member <strong>fragility</strong> curves for a five-storey <strong>frame</strong> system <strong>designed</strong> DC M <strong>and</strong> to PGA=0.20g <strong>and</strong>PGA=0.25g using IDA method for most critical storey members.91


Chapter 8: Results <strong>and</strong> Discussion0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1DC M DC HFigure 8.19 Member <strong>fragility</strong> curves for a five-storey <strong>frame</strong> system <strong>designed</strong> PGA=0.25g <strong>and</strong> to DC M <strong>and</strong>DC H using IDA method for most critical storey members.92


10.50Chapter 8: Results <strong>and</strong> 0 Discussion 0.2 0.4 0.6 0.8 11st 2nd 3rd 4th 5thFigure 8.20 Fragility curves <strong>of</strong> five-storey <strong>buildings</strong> <strong>designed</strong> to PGA=0.25g <strong>and</strong> DC M analyzed usingIDA method: (top) <strong>frame</strong> <strong>buildings</strong>; (bottom) <strong>frame</strong>-equivalent <strong>buildings</strong>93


10.50Chapter 8: Results <strong>and</strong> 0 Discussion 0.2 0.4 0.6 0.8 11st 2nd 3rd 4th 5thFigure 8.21 Fragility curves <strong>of</strong> five-storey <strong>buildings</strong> <strong>designed</strong> to PGA=0.25g <strong>and</strong> DC M analyzed usingIDA method: (top) <strong>frame</strong> <strong>buildings</strong>; (bottom) <strong>wall</strong>-equivalent <strong>buildings</strong>94


10.50Chapter 8: Results <strong>and</strong> 0 Discussion 0.2 0.4 0.6 0.8 11st 2nd 3rd 4th 5thFigure 8.22 Fragility curves <strong>of</strong> five-storey <strong>buildings</strong> <strong>designed</strong> to PGA=0.25g <strong>and</strong> DC M analyzed usingIDA method: (top) <strong>frame</strong> <strong>buildings</strong>; (bottom) <strong>wall</strong> <strong>buildings</strong>95


Chapter 8: Results <strong>and</strong> Discussion8.5. Comparison between analysis methodsAs described in previous chapters the methods <strong>of</strong> analysis performed for the construction <strong>of</strong>member <strong>fragility</strong> curves are the Incremental Dynamic Analysis (IDA) <strong>and</strong> the Static PushoverAnalysis (SPO). These <strong>fragility</strong> curves were then compared against results taken from asimplified analysis using the lateral force method (LFM) by Papailia [2011].Conclusions <strong>and</strong> observations about the comparison between each analysis method can bemade from the median PGA at the attainment <strong>of</strong> each damage state (see Table 8.5 to Table8.11) <strong>and</strong> the <strong>fragility</strong> curves as illustrated in Appendix B1 for all the examined <strong>buildings</strong>.Also Appendix B2 presents the <strong>fragility</strong> comparisons <strong>of</strong> the three methods for the mostcritical members.Comparing the three methods the following observations can be made:11Examples <strong>of</strong> beam <strong>fragility</strong> curves in the yielding damage state for the three methods(LFM, SPO <strong>and</strong> IDA) for the most critical members are presented in Figure 8.23 <strong>and</strong>0.5 Figure 8.24 for <strong>wall</strong>-<strong>frame</strong> 0.5 <strong>buildings</strong> <strong>and</strong> Figure 8.25 for <strong>frame</strong> <strong>buildings</strong>. It can beobserved that the three methods yield similar results.In five-storey <strong>buildings</strong> the <strong>fragility</strong> results <strong>of</strong> beams in the yielding damage statetaken 0 from LFM are slightly 0 higher than SPO <strong>and</strong> IDA (see Figure 8.23 <strong>and</strong> Figure0 0.2 0.4 0 0.6 0.20.8 0.4 1 0.61.2 0.81.41 1.2 1.48.25). In eight-storey <strong>buildings</strong> there is a larger difference between <strong>fragility</strong> resultstaken from LFM <strong>and</strong> the other two methods. (see Figure 8.24)LFM (ε>1) LFM (ε=1) LFM (ε>1) SPO LFM (ε=1) IDA SPO IDAFigure 8.23 Beam <strong>fragility</strong> curves in yielding state for five-storey <strong>frame</strong>-equivalent building <strong>designed</strong> toDC M <strong>and</strong> PGA=0.20g (left) <strong>and</strong> <strong>wall</strong>-equivalent building <strong>designed</strong> to DC H <strong>and</strong> PGA=0.25g (right).96


0.50.5Chapter 8: 0 Results <strong>and</strong> Discussion 00 0.2 0.4 0 0.6 0.20.8 0.4 1 0.61.2 0.81.41 1.2 1.4LFM (ε>1) LFM (ε=1) LFM (ε>1) SPO LFM (ε=1) IDA SPO IDA110.50.500Figure 8.24 0 Beam 0.2 <strong>fragility</strong> 0.4 curves in 0 0.6 yielding 0.2state 0.8 for 0.4 eight-storey 1 0.61.2 <strong>frame</strong>-equivalent 0.81.41building 1.2 <strong>designed</strong> 1.4 toDC M <strong>and</strong> PGA=0.20g (left) <strong>and</strong> <strong>wall</strong>-equivalent building <strong>designed</strong> to DC M <strong>and</strong> PGA=0.25g (right).LFM (ε>1) LFM (ε=1) LFM (ε>1) SPO LFM (ε=1) IDA SPO IDAFigure 8.25 Beam <strong>fragility</strong> curves in yielding state for five-storey <strong>frame</strong> building <strong>designed</strong> to PGA=0.25g<strong>and</strong> DC M (left) <strong>and</strong> DC H (right).The <strong>fragility</strong> curves <strong>of</strong> beams in the ultimate damage state, for the most criticalmembers, illustrate that the three methods have similar results (Figure 8.26 <strong>and</strong> Figure8.27 for <strong>wall</strong>-<strong>frame</strong> <strong>buildings</strong> <strong>and</strong> Figure 8.28 for <strong>frame</strong> <strong>buildings</strong>). The method withthe highest or lowest <strong>fragility</strong> is neither marked nor systematic.97


0.50.5Chapter 08: Results <strong>and</strong> Discussion 00 0.2 0.4 0 0.6 0.20.8 0.4 1 0.61.2 0.81.41 1.2 1.4LFM (ε>1) LFM (ε=1) LFM (ε>1) SPO LFM (ε=1) IDA SPO IDA110.50.500Figure 8.26 0 Beam 0.2<strong>fragility</strong> 0.4 curves 0in 0.6 ultimate 0.20.8 state for 0.4five-storey 1 0.61.2 <strong>frame</strong>-equivalent 0.81.41building 1.2 <strong>designed</strong> 1.4 toDC M <strong>and</strong> PGA=0.25g (left) <strong>and</strong> <strong>wall</strong>-equivalent building <strong>designed</strong> to DC M <strong>and</strong> PGA=0.20g (right).LFM (ε>1) LFM (ε=1) LFM (ε>1) SPO LFM (ε=1) IDA SPO IDA110.50.500Figure 8.27 0 Beam 0.2 <strong>fragility</strong> 0.4 curves in 0 0.6 ultimate 0.2state 0.8 for 0.4 eight-storey 1 0.61.2 <strong>frame</strong>-equivalent 0.81.41building 1.2 <strong>designed</strong> 1.4 toDC M <strong>and</strong> PGA=0.20g (left) <strong>and</strong> <strong>wall</strong>-equivalent building <strong>designed</strong> to DC M <strong>and</strong> PGA=0.25g (right).LFM (ε>1) LFM (ε=1) LFM (ε>1) SPO LFM (ε=1) IDA SPO IDAFigure 8.28 Beam <strong>fragility</strong> curves in ultimate state for five-storey <strong>frame</strong> building <strong>designed</strong> to PGA=0.25g98<strong>and</strong> DC M (left) <strong>and</strong> DC H (right).


Chapter 8: Results <strong>and</strong> Discussion11Examples <strong>of</strong> the <strong>fragility</strong> curves for the columns in yielding damage state for the most0.5 critical members taken 0.5 from results <strong>of</strong> the three methods are presented in Figure 8.29<strong>and</strong> Figure 8.30 for <strong>wall</strong>-<strong>frame</strong> <strong>buildings</strong> <strong>and</strong> in Figure 8.31 for <strong>frame</strong> <strong>buildings</strong>. Theresults obtained for the columns in yielding damage state using IDA <strong>and</strong> SPO in fivestorey0 <strong>buildings</strong> match well 0 for both <strong>frame</strong> <strong>and</strong> <strong>wall</strong>-<strong>frame</strong> <strong>buildings</strong> (see Figure 8.290 0.2 0.4 0 0.6 0.20.8 0.4 1 0.61.2 0.81.41 1.2 1.4<strong>and</strong> Figure 8.31). In eight-storey <strong>wall</strong>-<strong>frame</strong> <strong>buildings</strong> the fragilities obtained throughLFM have higher fragilities than the other two methods (see Figure 8.30).LFM (ε>1) LFM (ε=1) LFM (ε>1) SPO LFM (ε=1) IDA SPO IDA110.50.500Figure 08.29 Column 0.2 <strong>fragility</strong> 0.4 curves 0 0.6 in yielding 0.20.8 state 0.4 for 1five-storey 0.61.2 <strong>frame</strong>-equivalent 0.81.41 building 1.2 <strong>designed</strong> 1.4toDC M <strong>and</strong> PGA=0.25g (left) <strong>and</strong> <strong>wall</strong>-equivalent building <strong>designed</strong> to DC M <strong>and</strong> PGA=0.20g (right).LFM (ε>1) LFM (ε=1) LFM (ε>1) SPO LFM (ε=1) IDA SPO IDAFigure 8.30 Column <strong>fragility</strong> curves in yielding state for eight-storey <strong>frame</strong>-equivalent building <strong>designed</strong>to DC M <strong>and</strong> PGA=0.20g (left) <strong>and</strong> <strong>wall</strong> building <strong>designed</strong> to DC M <strong>and</strong> PGA=0.20g (right).99


0.50.5Chapter 0 8: Results <strong>and</strong> Discussion 00 0.2 0.4 0 0.6 0.20.8 0.4 1 0.61.2 0.81.41 1.2 1.4LFM (ε>1) LFM (ε=1) LFM (ε>1) SPO LFM (ε=1) IDA SPO IDAFigure 8.31 Column <strong>fragility</strong> curves in yielding state for five-storey <strong>frame</strong> building <strong>designed</strong> to DC M <strong>and</strong>PGA=0.20g <strong>and</strong> (left) PGA=0.25g (right). Examples <strong>of</strong> column <strong>fragility</strong> curves in ultimate damage state for the most criticalmembers are shown for the three analysis methods in Figure 8.32, Figure 8.33 <strong>and</strong>1Figure 8.34 for <strong>wall</strong>-<strong>frame</strong>1<strong>buildings</strong> <strong>and</strong> in Figure 8.35 in <strong>frame</strong> <strong>buildings</strong>. In fivestorey<strong>wall</strong>-<strong>frame</strong> <strong>buildings</strong> the <strong>fragility</strong> curves for the columns in their ultimatedamage state obtained using IDA <strong>and</strong> SPO match well, whereas the ones taken from0.5the LFM are slightly0.5lower (see Figure 8.32, Figure 8.33). In eight-storey <strong>wall</strong>-<strong>frame</strong><strong>buildings</strong> the three methods yield similar <strong>fragility</strong> results (see Figure 8.34). In fivestorey<strong>frame</strong> <strong>buildings</strong> the <strong>fragility</strong> curves for the columns in their ultimate damage0state obtained using IDA00 0.2 0.4 0 0.6 <strong>and</strong> LFM 0.20.8 match 0.4well, 1 whereas 0.61.2 0.8 the 1.4 ones taken 1 from 1.2 the SPO 1.4are slightly higher (see Figure 8.35).LFM (ε>1) LFM (ε=1) LFM (ε>1) SPO LFM (ε=1) IDA SPO IDAFigure 8.32 Column <strong>fragility</strong> curves in ultimate state for five-storey <strong>wall</strong> -equivalent building <strong>designed</strong> toDC M <strong>and</strong> PGA=0.25g (left) <strong>and</strong> <strong>wall</strong> building <strong>designed</strong> to DC M <strong>and</strong> PGA=0.25g (right).100


0.50.5Chapter 08: Results <strong>and</strong> Discussion 00 0.2 0.4 0 0.6 0.20.8 0.4 1 0.61.2 0.81.41 1.2 1.4LFM (ε>1) LFM (ε=1) LFM (ε>1) SPO LFM (ε=1) IDA SPO IDA110.50.500Figure 8.33 0 Column 0.2<strong>fragility</strong> 0.4curves 0in 0.6ultimate 0.20.8 state for 0.4five-storey 1 0.61.2 <strong>frame</strong>-equivalent 0.81.41 building 1.2<strong>designed</strong> 1.4 toDC H <strong>and</strong> PGA=0.25g (left) <strong>and</strong> <strong>wall</strong>-equivalent building <strong>designed</strong> to DC H <strong>and</strong> PGA=0.25g (right).LFM (ε>1) LFM (ε=1) LFM (ε>1) SPO LFM (ε=1) IDA SPO IDA110.50.500Figure 8.34 0 Column 0.2 <strong>fragility</strong> 0.4curves 0 0.6 in ultimate 0.20.8 state 0.4 for eight-storey 1 0.61.2 <strong>frame</strong>-equivalent 0.81.41 building 1.2 <strong>designed</strong> 1.4to DC M <strong>and</strong> PGA=0.25g (left) <strong>and</strong> <strong>wall</strong>-equivalent building <strong>designed</strong> to DC M <strong>and</strong> PGA=0.25g (right).LFM (ε>1) LFM (ε=1) LFM (ε>1) SPO LFM (ε=1) IDA SPO IDAFigure 8.35 Column <strong>fragility</strong> curves in ultimate state for five-storey <strong>frame</strong> building <strong>designed</strong> to DC M <strong>and</strong>101PGA=0.20g <strong>and</strong> (left) DC H <strong>and</strong> PGA=0.25g (right).


11Chapter 8: Results <strong>and</strong> Discussion0.50.5Fragility curves in <strong>wall</strong>-<strong>frame</strong> <strong>buildings</strong> for <strong>wall</strong>s in the yielding damage state takenfor 0 the three methods are 0 shown in Figure 8.36 <strong>and</strong> Figure 8.37. The three methods0 0.2 0.4 0 0.6 0.20.8 0.4 1 0.61.2 0.81.41 1.2 1.4yield similar results; the results taken from IDA have the lowest fragilities <strong>and</strong> theones taken from SPO <strong>and</strong> LFM match well. (see Figure 8.36 <strong>and</strong> Figure 8.37).LFM (ε>1) LFM (ε=1) LFM (ε>1) SPO LFM (ε=1) IDA SPO IDA110.50.500Figure 0 8.360.2 Wall <strong>fragility</strong> 0.4 curves 0 0.6in yielding 0.20.8 state 0.4for 1 five-storey 0.61.2 <strong>frame</strong>-equivalent 0.81.41 building 1.2 <strong>designed</strong> 1.4 toDC M <strong>and</strong> PGA=0.25g (left) <strong>and</strong> <strong>wall</strong> building <strong>designed</strong> to DC M <strong>and</strong> PGA=0.20g (right).LFM (ε>1) LFM (ε=1) LFM (ε>1) SPO LFM (ε=1) IDA SPO IDAFigure 8.37 Wall <strong>fragility</strong> curves in yielding state for five-storey <strong>frame</strong>-equivalent building <strong>designed</strong> toDC M <strong>and</strong> PGA=0.25g (left) <strong>and</strong> <strong>wall</strong> building <strong>designed</strong> to DC M <strong>and</strong> PGA=0.20g (right).Examples <strong>of</strong> <strong>fragility</strong> curves for the <strong>wall</strong>s in the ultimate damage state in flexure areillustrated in Figure 8.38 <strong>and</strong> Figure 8.39. The three methods yield similar results (seeFigure 8.38) except in five- <strong>and</strong> eight-storey <strong>wall</strong> <strong>dual</strong> systems where fragilitiesobtained from IDA are slightly higher. (see Figure 8.39).102


0.50.5Chapter 8: 0 Results <strong>and</strong> Discussion 00 0.2 0.4 0 0.6 0.20.8 0.4 1 0.61.2 0.81.41 1.2 1.4LFM (ε>1) LFM (ε=1) LFM (ε>1) SPO LFM (ε=1) IDA SPO IDA110.50.5Figure 8.38 Wall <strong>fragility</strong> curves in ultimate state in flexure for five-storey <strong>frame</strong>-equivalent building00<strong>designed</strong> to 0 DC M 0.2 <strong>and</strong> PGA=0.25g 0.4 (left) 0 0.6<strong>and</strong> 0.2 <strong>wall</strong>-equivalent 0.8 0.4 1 building 0.61.2 <strong>designed</strong> 0.81.4to DC 1H <strong>and</strong> PGA=0.25g 1.2 1.4(right).LFM (ε>1) LFM (ε=1) LFM (ε>1) SPO LFM (ε=1) IDA SPO IDAFigure 8.39 Wall <strong>fragility</strong> curves in ultimate state in flexure for five-storey <strong>wall</strong> building <strong>designed</strong> to DCM <strong>and</strong> PGA=0.25g (left) <strong>and</strong> eight-storey <strong>wall</strong> building <strong>designed</strong> to DC M <strong>and</strong> PGA=0.20g (right).Figure 8.40 to Figure 8.53 present the beam <strong>and</strong> column <strong>fragility</strong> curves for the three methodsin the yielding <strong>and</strong> ultimate damage state for both <strong>wall</strong>-<strong>frame</strong> <strong>and</strong> <strong>frame</strong> <strong>buildings</strong>. The firstcolumn in each set concerns the <strong>fragility</strong> curves obtained from IDA the second from SPO <strong>and</strong>the third are obtained from LFM. The first row in each set is for yielding <strong>and</strong> the second is forultimate damage state.The <strong>fragility</strong> results <strong>of</strong> beams in the yielding damage state show that the middle-storeybeams have the highest <strong>fragility</strong> whereas the top-storey beams have the lowest<strong>fragility</strong> for the three analysis methods in all the <strong>buildings</strong> examined (see Figure 8.40to Figure 8.45).The <strong>fragility</strong> results for the beams in the ultimate damage state show that middlestoreybeams have highest <strong>fragility</strong> for all three methods. The first-storey beams in theultimate damage states for IDA results <strong>and</strong> top-storey beams for SPO <strong>and</strong> LFM results103


Chapter 8: Results <strong>and</strong> Discussionhave the lowest fragilities. These observations hold for five- <strong>and</strong> eight-storey <strong>buildings</strong><strong>and</strong> for the three ranges <strong>of</strong> the ratio <strong>of</strong> total base shear taken by the <strong>wall</strong>s in <strong>wall</strong>-<strong>frame</strong><strong>buildings</strong> (see Figure 8.40 to Figure 8.45) <strong>and</strong> for all <strong>frame</strong> <strong>buildings</strong> (see Figure8.46).68 x 10-3 1st 2nd 3rd 4th 5th4200 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1Figure 8.40 Beam8 x <strong>fragility</strong> 10-3 curves for a) yielding <strong>and</strong> b) ultimate state for five-storey <strong>frame</strong>-equivalentbuilding <strong>designed</strong> to DC M <strong>and</strong> PGA=0.25g analyzed with IDA (left), SPO (middle) <strong>and</strong> LFM (right).61st 2nd 3rd 4th 5th4200 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1104Figure 8.41 Beam <strong>fragility</strong> curves for a) yielding <strong>and</strong> b) ultimate state for five-storey <strong>wall</strong>-equivalentbuilding <strong>designed</strong> to DC M <strong>and</strong> PGA=0.25g analyzed with IDA (left), SPO (middle) <strong>and</strong> LFM (right).


Chapter 8: Results <strong>and</strong> Discussion68 x 10-3 1st 2nd 3rd 4th 5th4200 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1Figure 8.42 Beam <strong>fragility</strong> curves for a) yielding <strong>and</strong> b) ultimate state for five-storey <strong>wall</strong> building0.4<strong>designed</strong> to DC M <strong>and</strong> PGA=0.25g analyzed with IDA (left), SPO (middle) <strong>and</strong> LFM (right).0.30.20.11st 2nd 3rd 4th5th 6th 7th 8th00 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1Figure 8.43 Beam <strong>fragility</strong> curves for a) yielding <strong>and</strong> b) ultimate state for eight-storey <strong>frame</strong>-equivalentbuilding <strong>designed</strong> to DC M <strong>and</strong> PGA=0.20g analyzed with IDA (left), SPO (middle) <strong>and</strong> LFM (right).105


Chapter 8: Results <strong>and</strong> Discussion0.40.30.21st 2nd 3rd 4th5th 6th 7th 8th0.100 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1Figure 8.44 Beam <strong>fragility</strong> curves for a) yielding <strong>and</strong> b) ultimate state for eight -storey <strong>wall</strong>-equivalent0.4building <strong>designed</strong> to DC M <strong>and</strong> PGA=0.20g analyzed with IDA (left), SPO (middle) <strong>and</strong> LFM (right).0.30.20.11st 2nd 3rd 4th5th 6th 7th 8th00 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1Figure 8.45 Beam <strong>fragility</strong> curves for a) yielding <strong>and</strong> b) ultimate state for eight -storey <strong>wall</strong> building<strong>designed</strong> to DC M <strong>and</strong> PGA=0.20g analyzed with IDA (left), SPO (middle) <strong>and</strong> LFM (right).106


Chapter 8: Results <strong>and</strong> Discussion68 x 10-3 1st 2nd 3rd 4th 5th4200 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1Figure 8.46 Beam <strong>fragility</strong> curves for a) yielding <strong>and</strong> b) ultimate state for five -storey <strong>frame</strong> building<strong>designed</strong> to DC M <strong>and</strong> PGA=0.20g analyzed with IDA (left), SPO (middle) <strong>and</strong> LFM (right).The <strong>fragility</strong> curves <strong>of</strong> columns in five-storey <strong>wall</strong>-<strong>frame</strong> <strong>and</strong> <strong>frame</strong> <strong>buildings</strong>, for theresults taken from IDA <strong>and</strong> SPO, show that the first-storey columns have the highest<strong>fragility</strong> <strong>and</strong> the top-storey columns have the lowest. The <strong>fragility</strong> curves for theresults taken from the LFM shows that the middle-storey columns have the highestfragilities <strong>and</strong> the top-storey columns the lowest. These observations hold for the threeranges <strong>of</strong> the ratio <strong>of</strong> total base shear taken by the <strong>wall</strong> (see Figure 8.47, Figure 8.48,Figure 8.49 for <strong>wall</strong>-<strong>frame</strong> building <strong>and</strong> Figure 8.53 for <strong>frame</strong> <strong>buildings</strong>).The <strong>fragility</strong> results <strong>of</strong> columns in eight-storey <strong>wall</strong>-<strong>frame</strong> <strong>buildings</strong> for the resultstaken from IDA <strong>and</strong> SPO show that the first- <strong>and</strong> top-storey columns have the highest<strong>fragility</strong> <strong>and</strong> the middle-storey columns have the lowest. The <strong>fragility</strong> curves <strong>of</strong> eightstorey<strong>buildings</strong> for columns in yielding damage state show that the results taken fromLFM are higher than IDA <strong>and</strong> SPO. These observations hold for all three ranges <strong>of</strong> theratio <strong>of</strong> total base shear taken by the <strong>wall</strong>s (see Figure 8.50, Figure 8.51 <strong>and</strong> Figure8.52).Fragility curves <strong>of</strong> columns for the non-critical members in <strong>wall</strong>-<strong>frame</strong> <strong>buildings</strong> forresults taken from LFM have higher fragilities than for the results taken from IDA <strong>and</strong>those taken from SPO are lower than the ones obtained from IDA. These observationshold for five- <strong>and</strong> eight- storey <strong>wall</strong>-<strong>frame</strong> <strong>buildings</strong> for all three ranges <strong>of</strong> the ratio <strong>of</strong>total base shear taken by the <strong>wall</strong>s (see Figure 8.47, Figure 8.48 <strong>and</strong> Figure 8.49)Fragilities <strong>of</strong> columns for the non-critical members in <strong>frame</strong> <strong>buildings</strong> for results takenfrom LFM have higher fragilities than for the results taken from SPO <strong>and</strong> those takenfrom SPO are higher than the ones obtained from IDA (see Figure 8.53).107


Chapter 8: Results <strong>and</strong> Discussion68 x 10-3 1st 2nd 3rd 4th 5th4200 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1Figure 8.47 Column 8 x <strong>fragility</strong> 10-3 curves for c) yielding <strong>and</strong> d) ultimate state for five-storey <strong>frame</strong>-equivalentbuilding <strong>designed</strong> to DC M <strong>and</strong> PGA=0.20g analyzed with IDA (left), SPO (middle) <strong>and</strong> LFM (right).61st 2nd 3rd 4th 5th4200 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1Figure 8.48 Column <strong>fragility</strong> curves for c) yielding <strong>and</strong> d) ultimate state for five-storey <strong>wall</strong>-equivalentbuilding <strong>designed</strong> to DC M <strong>and</strong> PGA=0.25g analyzed with IDA (left), SPO (middle) <strong>and</strong> LFM (right).108


Chapter 8: Results <strong>and</strong> Discussion68 x 10-3 1st 2nd 3rd 4th 5th4200 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1Figure 8.49 Column <strong>fragility</strong> curves for c) yielding <strong>and</strong> d) ultimate state for five-storey <strong>wall</strong> building0.4<strong>designed</strong> to DC M <strong>and</strong> PGA=0.25g analyzed with IDA (left), SPO (middle) <strong>and</strong> LFM (right).0.30.20.11st 2nd 3rd 4th5th 6th 7th 8th00 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1Figure 8.50 Column <strong>fragility</strong> curves for c) yielding <strong>and</strong> d) ultimate state for eight-storey <strong>frame</strong>-equivalentbuilding <strong>designed</strong> to DC M <strong>and</strong> PGA=0.20g analyzed with IDA (left), SPO (middle) <strong>and</strong> LFM (right).109


Chapter 8: Results <strong>and</strong> Discussion0.40.30.21st 2nd 3rd 4th5th 6th 7th 8th0.100 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1Figure 8.51 Column <strong>fragility</strong> curves for c) yielding <strong>and</strong> d) ultimate state for eight -storey <strong>wall</strong>-equivalent0.4building <strong>designed</strong> to DC M <strong>and</strong> PGA=0.20g analyzed with IDA (left), SPO (middle) <strong>and</strong> LFM (right).0.30.20.11st 2nd 3rd 4th5th 6th 7th 8th00 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1Figure 8.52 Column <strong>fragility</strong> curves for c) yielding <strong>and</strong> d) ultimate state for eight -storey <strong>wall</strong> building<strong>designed</strong> to DC M <strong>and</strong> PGA=0.20g analyzed with IDA (left), SPO (middle) <strong>and</strong> LFM (right).110


Chapter 8: Results <strong>and</strong> Discussion68 x 10-3 1st 2nd 3rd 4th 5th4200 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1Figure 8.53 Column <strong>fragility</strong> curves for c) yielding <strong>and</strong> d) ultimate state for five -storey <strong>frame</strong> building<strong>designed</strong> to DC M <strong>and</strong> PGA=0.20g analyzed with IDA (left), SPO (middle) <strong>and</strong> LFM (right).8.6. Fragility results <strong>of</strong> <strong>wall</strong>s in the ultimate stateAs previously explained the <strong>fragility</strong> curves are obtained from nonlinear dynamic analysis(IDA), nonlinear static analysis (SPO) <strong>and</strong> from a simplified analysis using the lateral forcemethod (LFM) by Papailia [2011]. The <strong>fragility</strong> curves <strong>of</strong> <strong>wall</strong>s for the ultimate damage statein shear obtained using the SPO is the envelope <strong>of</strong> the <strong>fragility</strong> curves for each storey. Highermode effects on <strong>wall</strong> shear dem<strong>and</strong>s are already taken into account in IDA.The shear dem<strong>and</strong>s obtained from the LFM are amplified by a factor ε (eq. 4.13) which takesinto account higher mode effects. This amplification is used for both DC M <strong>and</strong> DC H <strong>wall</strong>sfollowing a detailed procedure according to Keintzel [1990] also adopted in Eurocode 8[CEN, 2004b].The median PGAs at attainment <strong>of</strong> the ultimate damage state <strong>of</strong> <strong>wall</strong>s are presented in Table8.12 <strong>and</strong> Table 8.13. The ultimate damage state indicates the envelope <strong>of</strong> the shear <strong>and</strong>flexural failure for IDA <strong>and</strong> LFM; i.e. the lowest median PGA at attainment <strong>of</strong> the ultimatedamage state between flexure <strong>and</strong> shear failure. LFM results presented use inelasticamplification due to higher mode effects.111


Chapter 8: Results <strong>and</strong> DiscussionTable 8.12 Median PGA (g) at attainment <strong>of</strong> the ultimate damage state for <strong>wall</strong>s in 5-storey <strong>buildings</strong>IDA 0.38g 0.42g 0.33g0.25g M LFM 0.19g 0.18g 0.19gIDA 0.39g 0.37g 0.29g0.25g H LFM 0.38g 0.38g 0.51gIDA 0.47g 0.45g 0.57gTable 8.13 Median PGA (g) at attainment <strong>of</strong> the ultimate damage state for <strong>wall</strong>s in 8-storey <strong>buildings</strong>design PGA DC Analysis FrameequivalenequivalentWall-Wallmethodsystems0.20g M LFM 0.25g 0.19g 0.17gdesign PGA DC Analysis FrameequivalenequivalentWall-Wallmethodsystems0.20g M LFM 0.31g 0.18g 0.18gIDA 0.39g 0.53g 0.29g0.25g M LFM 0.33g 0.18g 0.19gIDA 0.40g 0.54g 0.35gThe conclusions on the <strong>wall</strong>s for the ultimate damage state (maximum <strong>of</strong> flexure <strong>and</strong> shearfailure) in <strong>wall</strong>-<strong>frame</strong> <strong>buildings</strong> are: Wall fragilities for the results obtained from LFM (where inelastic amplification for thehigher mode effects is used) show that the DC M <strong>wall</strong>s in <strong>wall</strong>-equivalent <strong>and</strong> <strong>wall</strong> <strong>dual</strong>systems may fail in shear before their design PGA. DC M <strong>wall</strong>s in <strong>frame</strong>-equivalentsystems fail beyond their design PGA in most cases. (see Table 8.12 <strong>and</strong> Table 8.13) Results taken from dynamic analysis (IDA) show that DC M <strong>wall</strong>s fail at PGA values 1.6to 1.9 times their design PGA in <strong>frame</strong>-equivalent systems, 1.5 to 2.5 times their designPGA in <strong>wall</strong>-equivalent systems <strong>and</strong> 1.2 to 1.4 times their design PGA in <strong>wall</strong> systems(see Table 8.12 <strong>and</strong> Table 8.13) The <strong>fragility</strong> results obtained for the DC H <strong>wall</strong>s show that for the results obtained fromthe LFM they fail at PGA values 1.5 to 2 times higher than their design PGA <strong>and</strong> 1.8 to2.2 times their design PGA for results obtained from IDA. (see Table 8.12 <strong>and</strong> Table8.13).The median PGAs at attainment <strong>of</strong> the shear failure <strong>of</strong> <strong>wall</strong>s are presented in Table 8.14 <strong>and</strong>Table 8.15. The values shown include the results obtained from IDA <strong>and</strong> the LFM. Theresults taken from the LFM are shown for both amplified LFM(ε>1) <strong>and</strong> non-amplifiedLFM(ε=1) shear dem<strong>and</strong>s.Examples <strong>of</strong> <strong>wall</strong> fragilities in the ultimate damage state in shear are presented in Figure 8.54,Figure 8.55 <strong>and</strong> Figure 8.56 for the results taken from (a) LFM with inelastic amplification <strong>of</strong>shear dem<strong>and</strong>s due to higher modes, b) LFM without the inelastic amplification <strong>of</strong> shear112


Chapter 8: Results <strong>and</strong> Discussiondem<strong>and</strong>s, c) incremental dynamic analysis (IDA) <strong>and</strong> d) the static pushover analysis (SPO).These figures are fully presented in Appendix A3 for all the examined <strong>buildings</strong>.The conclusions on <strong>wall</strong>s for the ultimate damage state in shear for <strong>wall</strong>-<strong>frame</strong> <strong>buildings</strong> are:The results taken from the dynamic analysis are in-between the LFM results with <strong>and</strong>without inelastic amplification <strong>of</strong> shear dem<strong>and</strong>s due to higher modes. (see Figure 8.54,Figure 8.55, Figure 8.56, Table 8.14 <strong>and</strong> Table 8.15). The latter applies for both DC M<strong>and</strong> DC H <strong>wall</strong>s, except in five- <strong>and</strong> eight-storey <strong>frame</strong>-equivalent <strong>buildings</strong> <strong>and</strong> eightstorey<strong>wall</strong>-equivalent systems (see Table 8.14 <strong>and</strong> Table 8.15)The dynamic analysis (IDA) confirms to a certain extent the inelastic amplification <strong>of</strong>shear forces due to higher modes in both DC M <strong>and</strong> DC H <strong>wall</strong>s <strong>and</strong> show that therelevant rules <strong>of</strong> Eurocode 8 are on the conservative side. The latter was also observed forDC H <strong>wall</strong>s in Ruttenberg <strong>and</strong> Nsieri [2006] <strong>and</strong> Kappos <strong>and</strong> Antoniadis [2007].The <strong>fragility</strong> curve results taken from SPO match well with the other two methods up toyielding. Beyond yielding there is no significant increase in shear force dem<strong>and</strong>s. (seeFigure 8.54, Figure 8.55 <strong>and</strong> Figure 8.56 ).Table 8.14 Median PGA (g) at attainment <strong>of</strong> the ultimate damage state in shear for <strong>wall</strong>s in 5-storey<strong>buildings</strong>IDA 0.94g 0.42g 0.33gLFM (ε=1) 0.41g 0.52g 0.54g0.25g M LFM (ε>1) 0.19g 0.18g 0.19gIDA 0.39g 0.37g 0.29gLFM (ε=1) 0.39g 0.52g 0.74g0.25g H LFM (ε>1) 0.38g 0.38g 0.51gIDA 0.90g 0.53g 0.72gLFM (ε=1) 0.61g 0.61g 0.98gTable 8.15 Median PGA (g) at attainment <strong>of</strong> the ultimate damage state in shear for <strong>wall</strong>s in 8-storey<strong>buildings</strong>design DC Analysis FrameequivalenequivalentWall-WallPGAmethodsystem0.20g M LFM (ε>1) 0.25g 0.19g 0.17gdesign DC Analysis FrameequivalenequivalentWall-WallPGAmethodsystem0.20g M LFM (ε>1) 0.58g 0.18g 0.18gIDA 0.88g 0.61g 0.29gLFM (ε=1) 0.58g 0.20g 0.44g0.25g M LFM (ε>1) 0.51g 0.18g 0.19gIDA 0.77g 0.67g 0.35gLFM (ε=1) 0.52g 0.20g 0.60g113


Chapter 8: Results <strong>and</strong> DiscussionFigure 8.54 Fragility curves <strong>of</strong> <strong>wall</strong>s for the ultimate damage state in shear <strong>of</strong> a five-storey <strong>wall</strong>-equivalentbuilding <strong>designed</strong> to DC M <strong>and</strong> PGA=0.20g.Figure 8.55 Fragility curves <strong>of</strong> <strong>wall</strong>s for the ultimate damage state in shear <strong>of</strong> a five-storey <strong>wall</strong> building<strong>designed</strong> to DC M <strong>and</strong> PGA=0.20g.114


Chapter 8: Results <strong>and</strong> DiscussionFigure 8.56 Fragility curves <strong>of</strong> <strong>wall</strong>s for the ultimate damage state in shear <strong>of</strong> a eight-storey <strong>wall</strong> building<strong>designed</strong> to DC M <strong>and</strong> PGA=0.20g.115


Chapter 8: Conclusions9. SUMMARY AND CONCLUSIONSThis study deals with the seismic <strong>fragility</strong> curves <strong>of</strong> reinforced concrete <strong>frame</strong> <strong>and</strong> <strong>wall</strong>-<strong>frame</strong>(<strong>dual</strong>) <strong>buildings</strong> <strong>designed</strong> according to Eurocode 2 <strong>and</strong> Eurocode 8 [CEN, 2004a,b]. Prototypeplan- <strong>and</strong> height- wise very regular <strong>buildings</strong> are studied with parameters including the height<strong>of</strong> the building, the level <strong>of</strong> Eurocode 8 design (in terms <strong>of</strong> design peak ground acceleration<strong>and</strong> ductility class) <strong>and</strong> for <strong>dual</strong> systems the percentage <strong>of</strong> seismic base shear taken by the<strong>wall</strong>s.The member fragilities were constructed using two different methods; incremental nonlineardynamic analysis (IDA) <strong>and</strong> nonlinear static (pushover) analysis (SPO). These methods wereperformed using a three-dimensional structural model <strong>of</strong> the full <strong>buildings</strong>. IDA is performedusing 14 spectrum-compatible semi-artificial accelerograms <strong>and</strong> SPO is performed usinginverted triangular load pattern. The N2 method is employed to combine the results <strong>of</strong> theSPO with the response spectrum analysis <strong>of</strong> an equivalent single degree-<strong>of</strong>-freedom system torelate the damage measure dem<strong>and</strong>s for each analysis step to the intensity measure (i.e. peakground acceleration). A simplified analysis using the lateral force method (LFM) by Papailia[2011] was compared against SPO <strong>and</strong> IDA.The results <strong>of</strong> the three analysis methods are presented in the form <strong>of</strong> <strong>fragility</strong> curves for twomember limit states; yielding <strong>and</strong> ultimate deformation in bending or shear. The structuraldamage <strong>of</strong> members is expressed in terms <strong>of</strong> peak ground acceleration (PGA) as the intensitymeasure (IM) since it is easier to compare it against the design PGA <strong>of</strong> the <strong>buildings</strong>. Thedamage measures (DM) taken are the peak chord rotation <strong>and</strong> the shear force dem<strong>and</strong>s atmember ends. The probability <strong>of</strong> exceedance <strong>of</strong> each limit state is computed from theprobability distributions <strong>of</strong> the damage measures (conditional on intensity measure) <strong>and</strong> <strong>of</strong> thecorresponding capacities.Dispersions for DM-dem<strong>and</strong>s are taken explicitly from the analysis for IDA method <strong>and</strong>estimates <strong>of</strong> dispersions <strong>of</strong> DM-dem<strong>and</strong>s are taken from previous studies for the SPO <strong>and</strong> theLFM. All three methods use estimates for the damage measure capacities based on previousstudies. It can be observed that the CoV-values determined through IDA are slightly lowerthan the ones determined from previous studies. The dispersions <strong>of</strong> DM-dem<strong>and</strong>s taken fromIDA for beams <strong>and</strong> columns have a larger scatter in the storeys in <strong>dual</strong> <strong>buildings</strong> than in <strong>frame</strong><strong>buildings</strong> <strong>and</strong> their mean is slightly higher.116


Chapter 8: ConclusionsThe results taken from Papailia [2011] using the LFM indicate that the <strong>wall</strong>s in <strong>buildings</strong><strong>designed</strong> according to Eurocode 8 <strong>and</strong> Medium Ductility Class are likely to fail in shear evenbefore the design PGA. The shear force dem<strong>and</strong>s taken from the LFM in concrete <strong>wall</strong>s areamplified to consider higher modes effects. Results from the nonlinear static (pushover) <strong>and</strong>dynamic analysis were used to better underst<strong>and</strong> the seismic behavior <strong>of</strong> regular <strong>dual</strong> or <strong>frame</strong><strong>buildings</strong> <strong>and</strong> the inelastic amplification <strong>of</strong> shear force dem<strong>and</strong>s in concrete <strong>wall</strong>s due tohigher modes.In <strong>wall</strong>-<strong>frame</strong> <strong>dual</strong> <strong>buildings</strong> the following conclusions <strong>and</strong> observations are made for theresults obtained from IDA <strong>and</strong> SPO:Walls are the most critical members for both yielding <strong>and</strong> ultimate damage states.Beams are much more likely to reach ultimate damage state than columns.Design to DC M in lieu <strong>of</strong> DC H may reduce the <strong>fragility</strong> <strong>of</strong> beams against yielding<strong>and</strong> increase <strong>fragility</strong> <strong>of</strong> columns. However, these effects are neither systematic notmarked. Wall <strong>fragility</strong> against yielding does not significantly change <strong>and</strong> is higher inDC M <strong>wall</strong>s in the ultimate damage state.Design to a higher PGA reduces the fragilities <strong>of</strong> beams <strong>and</strong> slightly increases those <strong>of</strong>columns. Wall fragilities do not significantly change.As the proportion <strong>of</strong> the total base shear taken by the <strong>wall</strong> increases the beams,columns <strong>and</strong> <strong>wall</strong>s in flexure have lower fragilities but <strong>wall</strong>s in shear ultimate statehave higher fragilities.Taller <strong>buildings</strong> generally exhibit lower fragilities for beams <strong>and</strong> columns <strong>and</strong> similarfragilities for <strong>wall</strong>s.The conclusions made on <strong>frame</strong> <strong>buildings</strong> for the results obtained from IDA <strong>and</strong> SPO are:Frame <strong>buildings</strong> give satisfactory <strong>fragility</strong> results even beyond their design PGA.Beams yield before their design PGA whereas the columns remain elastic well beyondthe design PGA.Beams are much more likely to reach yielding <strong>and</strong> collapse than in columns.Design to DC M instead <strong>of</strong> DC H may reduce slightly the <strong>fragility</strong> <strong>of</strong> beams <strong>and</strong>columns against yielding, but may increase that <strong>of</strong> beams against ultimateDesign for higher PGA reduces only slightly the fragilities <strong>of</strong> beams <strong>and</strong> columns inyielding state <strong>and</strong> may increase <strong>fragility</strong> in the ultimate state.Wall-<strong>frame</strong> (<strong>dual</strong>) systems have, in general, higher <strong>fragility</strong> than <strong>frame</strong> systems forcolumns <strong>and</strong> lower for beams.The conclusions when comparing the alternative analysis methods are:The alternative methods yield results that are in good agreement with either damagestate <strong>of</strong> columns <strong>and</strong> beams in both <strong>frame</strong> <strong>and</strong> <strong>dual</strong> <strong>buildings</strong> <strong>and</strong> to the flexuralbehavior <strong>of</strong> <strong>wall</strong>s. Larger differences are observed in eight-storey <strong>wall</strong>-<strong>frame</strong> <strong>dual</strong><strong>buildings</strong> for columns in yielding state where the <strong>fragility</strong> results taken from LFM arehigher than IDA <strong>and</strong> SPO.117


Chapter 8: ConclusionsWall fragilities for the results obtained from LFM (where inelastic amplification forthe higher mode effects is used) show that the DC M <strong>wall</strong>s in <strong>wall</strong>-equivalent <strong>and</strong> <strong>wall</strong><strong>dual</strong> systems may fail in shear before their design PGA. Walls in <strong>frame</strong>-equivalentsystems are likely to fail at PGA values beyond the design PGA.Results taken from dynamic analysis (IDA) show that DC M <strong>wall</strong>s fail at PGA values1.6 to 1.9 times their design PGA in <strong>frame</strong>-equivalent systems, 1.5 to 2.5 times theirdesign PGA in <strong>wall</strong>-equivalent systems <strong>and</strong> 1.2 to 1.4 times their design PGA in <strong>wall</strong>systems. The <strong>fragility</strong> results obtained for the DC H <strong>wall</strong>s show that they fail at PGA values 1.5to 2 times higher than the design PGA for the results obtained from the LFM <strong>and</strong> 1.8to 2.2 times the design PGA for results obtained from dynamic analysis (IDA).The <strong>fragility</strong> curve results for <strong>wall</strong> shear failure taken from SPO match well with theother two methods up to yielding. Beyond yielding there is no significant increase inshear force dem<strong>and</strong>s.The results taken from the dynamic analysis (IDA) are in-between the LFM resultswith <strong>and</strong> without inelastic amplification <strong>of</strong> shear dem<strong>and</strong>s due to higher modes. Thelatter applies in both DC M <strong>and</strong> DC H <strong>wall</strong>s, except in five- <strong>and</strong> eight-storey <strong>frame</strong>equivalent<strong>buildings</strong> <strong>and</strong> eight-storey <strong>wall</strong>-equivalent systems.The dynamic analysis confirm to a certain extent the inelastic amplification <strong>of</strong> shearforces due to higher modes in both DC M <strong>and</strong> DC H <strong>wall</strong>s <strong>and</strong> show that the relevantrules <strong>of</strong> Eurocode 8 are on the conservative side. The latter was also observed for DCH <strong>wall</strong>s in Ruttenberg <strong>and</strong> Nsieri [2006] <strong>and</strong> Kappos <strong>and</strong> Antoniadis [2007].118


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AppendicesAPPENDIX AA1. Fragility curves using Incremental Dynamic analysisAppendix A1 presents the member <strong>fragility</strong> curves <strong>of</strong> each examined <strong>buildings</strong> for the twodamage states <strong>of</strong> yielding <strong>and</strong> ultimate for the analysis performed using Incremental dynamicanalysis (IDA). The sub-figures (a) <strong>and</strong> (b) refer to the beam yielding <strong>and</strong> ultimate damagestate in flexure, (c) <strong>and</strong> (d) refer to column yielding <strong>and</strong> ultimate damage state in flexure. In<strong>wall</strong>-<strong>frame</strong> <strong>dual</strong> <strong>buildings</strong> (e) <strong>and</strong> (f) refer to <strong>wall</strong> yielding <strong>and</strong> ultimate damage state inflexure <strong>and</strong> (g) refers to <strong>wall</strong> ultimate damage state in shear. In <strong>frame</strong> <strong>buildings</strong> (e) <strong>and</strong> (f)refer to beam <strong>and</strong> column ultimate damage state in shear respectively.A1


Appendices10.500 0.2 0.4 0.6 0.8 11st 2nd 3rd 4th 5thFigure A. 1 Fragility curves <strong>of</strong> five-storey <strong>frame</strong> building <strong>designed</strong> to DC M <strong>and</strong> PGA=0.20g using IDAA2


Appendices10.500 0.2 0.4 0.6 0.8 11st 2nd 3rd 4th 5thFigure A. 2 Fragility curves <strong>of</strong> five-storey <strong>frame</strong> building <strong>designed</strong> to DC M <strong>and</strong> PGA=0.25g using IDAA3


Appendices10.500 0.2 0.4 0.6 0.8 11st 2nd 3rd 4th 5thFigure A. 3 Fragility curves <strong>of</strong> five-storey <strong>frame</strong> building <strong>designed</strong> to DC H <strong>and</strong> PGA=0.25g using IDAA4


Appendices10.500 0.2 0.4 0.6 0.8 11st 2nd 3rd 4th 5thFigure A. 4 Fragility curves <strong>of</strong> five-storey <strong>frame</strong>-equivalent building <strong>designed</strong> to DC M <strong>and</strong> PGA=0.20gusing IDAA5


Appendices10.500 0.2 0.4 0.6 0.8 11st 2nd 3rd 4th 5thFigure A. 5 Fragility curves <strong>of</strong> five-storey <strong>frame</strong>-equivalent building <strong>designed</strong> to DC M <strong>and</strong> PGA=0.25gusing IDAA6


Appendices10.500 0.2 0.4 0.6 0.8 11st 2nd 3rd 4th 5thFigure A. 6 Fragility curves <strong>of</strong> five-storey <strong>frame</strong>-equivalent building <strong>designed</strong> to DC H <strong>and</strong> PGA=0.25gusing IDAA7


Appendices10.500 0.2 0.4 0.6 0.8 11st 2nd 3rd 4th 5thFigure A. 7 Fragility curves <strong>of</strong> five-storey <strong>wall</strong>-equivalent building <strong>designed</strong> to DC M <strong>and</strong> PGA=0.20gusing IDAA8


Appendices10.500 0.2 0.4 0.6 0.8 11st 2nd 3rd 4th 5thFigure A. 8 Fragility curves <strong>of</strong> five-storey <strong>wall</strong>-equivalent building <strong>designed</strong> to DC M <strong>and</strong> PGA=0.25gusing IDAA9


Appendices10.500 0.2 0.4 0.6 0.8 11st 2nd 3rd 4th 5thFigure A. 9 Fragility curves <strong>of</strong> five-storey <strong>wall</strong>-equivalent building <strong>designed</strong> to DC H <strong>and</strong> PGA=0.25gusing IDAA10


Appendices10.500 0.2 0.4 0.6 0.8 11st 2nd 3rd 4th 5thFigure A. 10 Fragility curves <strong>of</strong> five-storey <strong>wall</strong> building <strong>designed</strong> to DC M <strong>and</strong> PGA=0.20g using IDAA11


Appendices10.500 0.2 0.4 0.6 0.8 11st 2nd 3rd 4th 5thFigure A. 11 Fragility curves <strong>of</strong> five-storey <strong>wall</strong> building <strong>designed</strong> to DC M <strong>and</strong> PGA=0.25g using IDAA12


Appendices10.500 0.2 0.4 0.6 0.8 11st 2nd 3rd 4th 5thFigure A. 12 Fragility curves <strong>of</strong> five-storey <strong>wall</strong> building <strong>designed</strong> to DC H <strong>and</strong> PGA=0.25g using IDAA13


Appendices10.51st 2nd 3rd 4th00 5th 0.2 6th 0.4 0.6 7th 0.8 8th1A14Figure A. 13 Fragility curves <strong>of</strong> eight-storey <strong>frame</strong>-equivalent <strong>dual</strong> system <strong>designed</strong> to DC M <strong>and</strong>PGA=0.20g using IDA


Appendices10.51st 2nd 3rd 4th00 5th 0.2 6th 0.4 0.6 7th 0.8 8th1A15Figure A. 14 Fragility curves <strong>of</strong> eight-storey <strong>frame</strong>-equivalent <strong>dual</strong> system <strong>designed</strong> to DC M <strong>and</strong>PGA=0.25g using IDA


Appendices10.51st 2nd 3rd 4th00 5th 0.2 6th 0.4 0.6 7th 0.8 8th1Figure A. 15 Fragility curves <strong>of</strong> eight-storey <strong>wall</strong>-equivalent <strong>dual</strong> system <strong>designed</strong> to DC M <strong>and</strong>PGA=0.20g using IDAA16


Appendices10.51st 2nd 3rd 4th00 5th 0.2 6th 0.4 0.6 7th 0.8 8th1Figure A. 16 Fragility curves <strong>of</strong> eight-storey <strong>wall</strong>-equivalent <strong>dual</strong> system <strong>designed</strong> to DC M <strong>and</strong>PGA=0.25g using IDAA17


Appendices10.51st 2nd 3rd 4th00 5th 0.2 6th 0.4 0.6 7th 0.8 8th1Figure A. 17 Fragility curves <strong>of</strong> eight-storey <strong>wall</strong> building <strong>designed</strong> to DC M <strong>and</strong> PGA=0.20g using IDAA18


Appendices10.51st 2nd 3rd 4th00 5th 0.2 6th 0.4 0.6 7th 0.8 8th1Figure A. 18 Fragility curves <strong>of</strong> eight-storey <strong>wall</strong> building <strong>designed</strong> to DC M <strong>and</strong> PGA=0.25g using IDAA19


AppendicesA2. Fragility curves using Static Pushover AnalysisAppendix A2 presents the member <strong>fragility</strong> curves <strong>of</strong> each examined building for the twodamage states <strong>of</strong> yielding <strong>and</strong> ultimate for the analysis performed using nonlinear static(pushover) analysis (SPO). The sub-figures (a) <strong>and</strong> (b) refer to the beam yielding <strong>and</strong> ultimatedamage state in flexure, (c) <strong>and</strong> (d) refer to column yielding <strong>and</strong> ultimate damage state inflexure. In <strong>wall</strong>-<strong>frame</strong> <strong>dual</strong> <strong>buildings</strong> (e) <strong>and</strong> (f) refer to <strong>wall</strong> yielding <strong>and</strong> ultimate damagestate in flexure <strong>and</strong> (g) refers to <strong>wall</strong> ultimate damage state in shear. In <strong>frame</strong> <strong>buildings</strong> (e)<strong>and</strong> (f) refer to beam <strong>and</strong> column ultimate damage state in shear respectively.A20


Appendices10.500 0.2 0.4 0.6 0.8 11st 2nd 3rd 4th 5thFigure A. 19 Fragility curves <strong>of</strong> five-storey <strong>frame</strong> building <strong>designed</strong> to DC M <strong>and</strong> PGA=0.20g using SPOA21


Appendices10.500 0.2 0.4 0.6 0.8 11st 2nd 3rd 4th 5thFigure A. 20 Fragility curves <strong>of</strong> five-storey <strong>frame</strong> building <strong>designed</strong> to DC M <strong>and</strong> PGA=0.25g using SPOA22


Appendices10.500 0.2 0.4 0.6 0.8 11st 2nd 3rd 4th 5thFigure A. 21 Fragility curves <strong>of</strong> five-storey <strong>frame</strong> building <strong>designed</strong> to DC H <strong>and</strong> PGA=0.25g using SPOA23


Appendices10.500 0.2 0.4 0.6 0.8 11st 2nd 3rd 4th 5thFigure A. 22 Fragility curves <strong>of</strong> five-storey <strong>frame</strong>-equivalent building <strong>designed</strong> to DC M <strong>and</strong> PGA=0.20gusing SPOA24


Appendices10.500 0.2 0.4 0.6 0.8 11st 2nd 3rd 4th 5thFigure A. 23 Fragility curves <strong>of</strong> five-storey <strong>frame</strong>-equivalent building <strong>designed</strong> to DC M <strong>and</strong> PGA=0.25gusing SPOA25


Appendices10.500 0.2 0.4 0.6 0.8 11st 2nd 3rd 4th 5thFigure A. 24 Fragility curves <strong>of</strong> five-storey <strong>frame</strong>-equivalent building <strong>designed</strong> to DC H <strong>and</strong> PGA=0.25gusing SPOA26


Appendices10.500 0.2 0.4 0.6 0.8 11st 2nd 3rd 4th 5thFigure A. 25 Fragility curves <strong>of</strong> five-storey <strong>wall</strong>-equivalent building <strong>designed</strong> to DC M <strong>and</strong> PGA=0.20gusing SPOA27


Appendices10.500 0.2 0.4 0.6 0.8 11st 2nd 3rd 4th 5thFigure A. 26 Fragility curves <strong>of</strong> five-storey <strong>wall</strong>-equivalent building <strong>designed</strong> to DC M <strong>and</strong> PGA=0.25gusing SPOA28


Appendices10.500 0.2 0.4 0.6 0.8 11st 2nd 3rd 4th 5thFigure A. 27 Fragility curves <strong>of</strong> five-storey <strong>wall</strong>-equivalent building <strong>designed</strong> to DC H <strong>and</strong> PGA=0.25gusing SPOA29


Appendices10.500 0.2 0.4 0.6 0.8 11st 2nd 3rd 4th 5thFigure A. 28 Fragility curves <strong>of</strong> five-storey <strong>wall</strong> building <strong>designed</strong> to DC M <strong>and</strong> PGA=0.20g using SPOA30


Appendices10.500 0.2 0.4 0.6 0.8 11st 2nd 3rd 4th 5thFigure A. 29 Fragility curves <strong>of</strong> five-storey <strong>wall</strong> building <strong>designed</strong> to DC M <strong>and</strong> PGA=0.25g using SPOA31


Appendices10.500 0.2 0.4 0.6 0.8 11st 2nd 3rd 4th 5thFigure A. 30 Fragility curves <strong>of</strong> five-storey <strong>wall</strong> building <strong>designed</strong> to DC H <strong>and</strong> PGA=0.25g using SPOA32


Appendices10.51st 2nd 3rd 4th00 5th 0.2 6th 0.4 0.6 7th 0.8 8th1A33Figure A. 31 Fragility curves <strong>of</strong> eight-storey <strong>frame</strong>-equivalent <strong>dual</strong> system <strong>designed</strong> to DC M <strong>and</strong>PGA=0.20g using SPO


Appendices10.51st 2nd 3rd 4th00 5th 0.2 6th 0.4 0.6 7th 0.8 8th1A34Figure A. 32 Fragility curves <strong>of</strong> eight-storey <strong>frame</strong>-equivalent <strong>dual</strong> system <strong>designed</strong> to DC M <strong>and</strong>PGA=0.25g using SPO


Appendices10.51st 2nd 3rd 4th00 5th 0.2 6th 0.4 0.6 7th 0.8 8th1Figure A. 33 Fragility curves <strong>of</strong> eight-storey <strong>wall</strong>-equivalent <strong>dual</strong> system <strong>designed</strong> to DC M <strong>and</strong>PGA=0.20g using SPOA35


Appendices10.51st 2nd 3rd 4th00 5th 0.2 6th 0.4 0.6 7th 0.8 8th1Figure A. 34 Fragility curves <strong>of</strong> eight-storey <strong>wall</strong>-equivalent <strong>dual</strong> system <strong>designed</strong> to DC M <strong>and</strong>PGA=0.25g using SPOA36


Appendices10.51st 2nd 3rd 4th00 5th 0.2 6th 0.4 0.6 7th 0.8 8th1Figure A. 35 Fragility curves <strong>of</strong> eight-storey <strong>wall</strong> building <strong>designed</strong> to DC M <strong>and</strong> PGA=0.20g using SPOA37


Appendices10.51st 2nd 3rd 4th00 5th 0.2 6th 0.4 0.6 7th 0.8 8th1Figure A. 36 Fragility curves <strong>of</strong> eight-storey <strong>wall</strong> building <strong>designed</strong> to DC M <strong>and</strong> PGA=0.25g using SPOA38


AppendicesA3. Fragility curves <strong>of</strong> <strong>wall</strong>s in shearAppendix A3 presents the <strong>fragility</strong> curves <strong>of</strong> <strong>wall</strong>s for the ultimate damage state in shearusing the alternative methods <strong>of</strong> analysis. The sub-figures (a) <strong>and</strong> (b) refer to the lateral forcemethod with <strong>and</strong> without inelastic amplification respectively which takes into account thehigher mode effects. (ε=1) indicates that the shear dem<strong>and</strong> taken from LFM is not amplifiedfor higher mode effects <strong>and</strong> (ε>1) indicates that the shear dem<strong>and</strong>s taken from LFM areamplified for higher mode effects. (c) <strong>and</strong> (d) refer to the incremental dynamic analysis (IDA)<strong>and</strong> the static pushover analysis (SPO) respectively.Figure A. 37 Fragility curves <strong>of</strong> <strong>wall</strong>s for the ultimate damage state in shear <strong>of</strong> a five-storey <strong>frame</strong>equivalentbuilding <strong>designed</strong> to DC M <strong>and</strong> PGA=0.20g.A39


AppendicesFigure A. 38 Fragility curves <strong>of</strong> <strong>wall</strong>s for the ultimate damage state in shear <strong>of</strong> a five-storey <strong>frame</strong>equivalentbuilding <strong>designed</strong> to DC M <strong>and</strong> PGA=0.25g.Figure A. 39 Fragility curves <strong>of</strong> <strong>wall</strong>s for the ultimate damage state in shear <strong>of</strong> a five-storey <strong>frame</strong>equivalentbuilding <strong>designed</strong> to DC H <strong>and</strong> PGA=0.25g.A40


AppendicesFigure A. 40 Fragility curves <strong>of</strong> <strong>wall</strong>s for the ultimate damage state in shear <strong>of</strong> a five-storey <strong>wall</strong>equivalentbuilding <strong>designed</strong> to DC M <strong>and</strong> PGA=0.20g.Figure A. 41 Fragility curves <strong>of</strong> <strong>wall</strong>s for the ultimate damage state in shear <strong>of</strong> a five-storey <strong>wall</strong>equivalentbuilding <strong>designed</strong> to DC M <strong>and</strong> PGA=0.25g.A41


AppendicesFigure A. 42 Fragility curves <strong>of</strong> <strong>wall</strong>s for the ultimate damage state in shear <strong>of</strong> a five-storey <strong>wall</strong>equivalentbuilding <strong>designed</strong> to DC M <strong>and</strong> PGA=0.25g.Figure A. 43 Fragility curves <strong>of</strong> <strong>wall</strong>s for the ultimate damage state in shear <strong>of</strong> a five-storey <strong>wall</strong> building<strong>designed</strong> to DC M <strong>and</strong> PGA=0.20g.A42


AppendicesFigure A. 44 Fragility curves <strong>of</strong> <strong>wall</strong>s for the ultimate damage state in shear <strong>of</strong> a five-storey <strong>wall</strong> building<strong>designed</strong> to DC M <strong>and</strong> PGA=0.25g.Figure A. 45 Fragility curves <strong>of</strong> <strong>wall</strong>s for the ultimate damage state in shear <strong>of</strong> a five-storey <strong>wall</strong> building<strong>designed</strong> to DC H <strong>and</strong> PGA=0.25g.A43


AppendicesFigure A. 46 Fragility curves <strong>of</strong> <strong>wall</strong>s for the ultimate damage state in shear <strong>of</strong> a eight-storey <strong>frame</strong>equivalentbuilding <strong>designed</strong> to DC M <strong>and</strong> PGA=0.20g.Figure A. 47 Fragility curves <strong>of</strong> <strong>wall</strong>s for the ultimate damage state in shear <strong>of</strong> a eight-storey <strong>frame</strong>equivalentbuilding <strong>designed</strong> to DC M <strong>and</strong> PGA=0.25g.A44


AppendicesFigure A. 48 Fragility curves <strong>of</strong> <strong>wall</strong>s for the ultimate damage state in shear <strong>of</strong> a eight-storey <strong>wall</strong>equivalentbuilding <strong>designed</strong> to DC M <strong>and</strong> PGA=0.20g.Figure A. 49 Fragility curves <strong>of</strong> <strong>wall</strong>s for the ultimate damage state in shear <strong>of</strong> a eight-storey <strong>wall</strong>equivalentbuilding <strong>designed</strong> to DC M <strong>and</strong> PGA=0.25g.A45


AppendicesFigure A. 50 Fragility curves <strong>of</strong> <strong>wall</strong>s for the ultimate damage state in shear <strong>of</strong> a eight-storey <strong>wall</strong>building <strong>designed</strong> to DC M <strong>and</strong> PGA=0.20g.Figure A. 51 Fragility curves <strong>of</strong> <strong>wall</strong>s for the ultimate damage state in shear <strong>of</strong> a eight-storey <strong>wall</strong>building <strong>designed</strong> to DC M <strong>and</strong> PGA=0.25g.A46


AppendicesAPPENDIX BB1. Fragility curves - comparison <strong>of</strong> three methodsAppendix B1 presents the <strong>fragility</strong> curves <strong>of</strong> members for the three methods <strong>of</strong> analysis. Themethods <strong>of</strong> analysis include 1) the Incremental Dynamic Analysis, 2) Static PushoverAnalysis <strong>and</strong> 3) the lateral force method. The plots illustrate the two damage states <strong>of</strong> yielding<strong>and</strong> ultimate. The sub-figures (a) <strong>and</strong> (b) refer to the beam yielding <strong>and</strong> ultimate state (c) <strong>and</strong>(d) refer to column yielding <strong>and</strong> ultimate state. In <strong>wall</strong>-<strong>frame</strong> systems (e) <strong>and</strong> (f) refer to <strong>wall</strong>yielding <strong>and</strong> ultimate state respectively. The ultimate state for all members is the envelope <strong>of</strong>the ultimate damage state in shear <strong>and</strong> in flexure.B1


Appendices1) Incremental Dynamic Analysis 468 x 10-3 1st 2nd 3rd 4th 5th200 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.92) Static Pushover Analysis68 x 10-3 1st 2nd 3rd 4th 5th4200 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.968 x 10-3 1st 2nd 3rd 4th 5th3) Lateral Force Method4200 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9Figure B. 1 Coefficient Fragility curves <strong>of</strong> five-storey <strong>frame</strong> systems <strong>designed</strong> to DC M <strong>and</strong> PGA=0.20gusing IDA, SPO <strong>and</strong> LFM analysisB2


Appendices1) Incremental Dynamic Analysis 468 x 10-3 1st 2nd 3rd 4th 5th200 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.92) Static Pushover Analysis68 x 10-3 1st 2nd 3rd 4th 5th4200 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.968 x 10-3 1st 2nd 3rd 4th 5th3) Lateral Force Method4200 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9Figure B. 2 Fragility curves <strong>of</strong> five-storey <strong>frame</strong> systems <strong>designed</strong> to DC M <strong>and</strong> PGA=0.25g using IDA,SPO <strong>and</strong> LFM analysisB3


Appendices1) Incremental Dynamic Analysis 468 x 10-3 1st 2nd 3rd 4th 5th200 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.92) Static Pushover Analysis68 x 10-3 1st 2nd 3rd 4th 5th4200 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.968 x 10-3 1st 2nd 3rd 4th 5th3) Lateral Force Method4200 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9Figure B. 3 Fragility curves <strong>of</strong> five-storey <strong>frame</strong> systems <strong>designed</strong> to DC H <strong>and</strong> PGA=0.25g using IDA,SPO <strong>and</strong> LFM analysisB4


Appendices1) Incremental Dynamic Analysis 468 x 10-3 1st 2nd 3rd 4th 5th200 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.92) Static Pushover Analysis68 x 10-3 1st 2nd 3rd 4th 5th4200 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 168 x 10-3 1st 2nd 3rd 4th 5th3) Lateral Force Method4200 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1Figure B. 4 Fragility curves <strong>of</strong> five-storey <strong>frame</strong>-equivalent <strong>dual</strong> systems <strong>designed</strong> to DC M <strong>and</strong>PGA=0.20g using IDA, SPO <strong>and</strong> LFM analysisB5


Appendices1) Incremental Dynamic Analysis 468 x 10-3 1st 2nd 3rd 4th 5th200 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.92) Static Pushover Analysis68 x 10-3 1st 2nd 3rd 4th 5th4200 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 168 x 10-3 1st 2nd 3rd 4th 5th3) Lateral Force Method4200 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1Figure B. 5 Fragility curves <strong>of</strong> five-storey <strong>frame</strong>-equivalent <strong>dual</strong> systems <strong>designed</strong> to DC M <strong>and</strong>PGA=0.25g using IDA, SPO <strong>and</strong> LFM analysisB6


Appendices1) Incremental Dynamic Analysis 468 x 10-3 1st 2nd 3rd 4th 5th200 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.92) Static Pushover Analysis68 x 10-3 1st 2nd 3rd 4th 5th4200 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 168 x 10-3 1st 2nd 3rd 4th 5th3) Lateral Force Method4200 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1Figure B. 6 Fragility curves <strong>of</strong> five-storey <strong>frame</strong>-equivalent <strong>dual</strong> systems <strong>designed</strong> to DC H <strong>and</strong>PGA=0.25g using IDA, SPO <strong>and</strong> LFM analysisB7


Appendices1) Incremental Dynamic Analysis 468 x 10-3 1st 2nd 3rd 4th 5th200 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.92) Static Pushover Analysis68 x 10-3 1st 2nd 3rd 4th 5th4200 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 168 x 10-3 1st 2nd 3rd 4th 5th3) Lateral Force Method4200 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1Figure B. 7 Fragility curves <strong>of</strong> five-storey <strong>wall</strong>-equivalent <strong>dual</strong> systems <strong>designed</strong> to DC M <strong>and</strong> PGA=0.20gusing IDA, SPO <strong>and</strong> LFM analysisB8


Appendices1) Incremental Dynamic Analysis 468 x 10-3 1st 2nd 3rd 4th 5th200 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.92) Static Pushover Analysis68 x 10-3 1st 2nd 3rd 4th 5th4200 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 168 x 10-3 1st 2nd 3rd 4th 5th3) Lateral Force Method4200 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1Figure B. 8 Fragility curves <strong>of</strong> five-storey <strong>wall</strong>-equivalent <strong>dual</strong> systems <strong>designed</strong> to DC M <strong>and</strong> PGA=0.25gusing IDA, SPO <strong>and</strong> LFM analysisB9


Appendices1) Incremental Dynamic Analysis 468 x 10-3 1st 2nd 3rd 4th 5th200 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.92) Static Pushover Analysis68 x 10-3 1st 2nd 3rd 4th 5th4200 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 168 x 10-3 1st 2nd 3rd 4th 5th3) Lateral Force Method4200 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1Figure B. 9 Fragility curves <strong>of</strong> five-storey <strong>wall</strong>-equivalent <strong>dual</strong> systems <strong>designed</strong> to DC H <strong>and</strong> PGA=0.25gusing IDA, SPO <strong>and</strong> LFM analysisB10


Appendices1) Incremental Dynamic Analysis 468 x 10-3 1st 2nd 3rd 4th 5th200 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.92) Static Pushover Analysis68 x 10-3 1st 2nd 3rd 4th 5th4200 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 168 x 10-3 1st 2nd 3rd 4th 5th3) Lateral Force Method4200 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1Figure B. 10 Fragility curves <strong>of</strong> five – storey <strong>wall</strong> <strong>dual</strong> systems <strong>designed</strong> to DC M <strong>and</strong> PGA=0.20g usingIDA, SPO <strong>and</strong> LFM analysisB11


Appendices1) Incremental Dynamic Analysis 468 x 10-3 1st 2nd 3rd 4th 5th200 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.92) Static Pushover Analysis68 x 10-3 1st 2nd 3rd 4th 5th4200 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 168 x 10-3 1st 2nd 3rd 4th 5th3) Lateral Force Method4200 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1Figure B. 11 Fragility curves <strong>of</strong> five-storey <strong>wall</strong> systems <strong>designed</strong> to DC M <strong>and</strong> PGA=0.25g using IDA,SPO <strong>and</strong> LFM analysisB12


Appendices1) Incremental Dynamic Analysis 468 x 10-3 1st 2nd 3rd 4th 5th200 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.92) Static Pushover Analysis68 x 10-3 1st 2nd 3rd 4th 5th4200 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 168 x 10-3 1st 2nd 3rd 4th 5th3) Lateral Force Method4200 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1Figure B. 12 Fragility curves <strong>of</strong> five-storey <strong>wall</strong> systems <strong>designed</strong> to DC H <strong>and</strong> PGA=0.25g for IDA, SPO<strong>and</strong> LFM analysisB13


Appendices0.41) Incremental Dynamic Analysis0.30.21st 2nd 3rd 4th5th 6th 7th 8th0.100 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10.42) Static Pushover Analysis0.30.20.11st 2nd 3rd 4th5th 6th 7th 8th00 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10.43) Lateral Force Method0.30.20.11st 2nd 3rd 4th5th 6th 7th 8th00 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1B14Figure B. 13 Fragility curves <strong>of</strong> eight-storey <strong>frame</strong>-equivalent <strong>dual</strong> system <strong>designed</strong> to DC M <strong>and</strong>PGA=0.20g for IDA, SPO <strong>and</strong> LFM analysis


Appendices0.41) Incremental Dynamic Analysis0.30.21st 2nd 3rd 4th5th 6th 7th 8th0.100 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10.42) Static Pushover Analysis0.30.20.11st 2nd 3rd 4th5th 6th 7th 8th00 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10.43) Lateral Force Method0.30.20.11st 2nd 3rd 4th5th 6th 7th 8th00 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1B15Figure B. 14 Fragility curves <strong>of</strong> eight-storey <strong>frame</strong>-equivalent <strong>dual</strong> system <strong>designed</strong> to DC M <strong>and</strong>PGA=0.25g for IDA, SPO <strong>and</strong> LFM analysis


Appendices0.41) Incremental Dynamic Analysis0.30.21st 2nd 3rd 4th5th 6th 7th 8th0.100 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10.42) Static Pushover Analysis0.30.20.11st 2nd 3rd 4th5th 6th 7th 8th00 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10.43) Lateral Force Method0.30.20.11st 2nd 3rd 4th5th 6th 7th 8th00 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1B16Figure B. 15 Fragility curves <strong>of</strong> eight-storey <strong>wall</strong>-equivalent <strong>dual</strong> systems <strong>designed</strong> to DC M <strong>and</strong>PGA=0.20g for IDA, SPO <strong>and</strong> LFM analysis


Appendices0.41) Incremental Dynamic Analysis0.30.21st 2nd 3rd 4th5th 6th 7th 8th0.100 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10.42) Static Pushover Analysis0.30.20.11st 2nd 3rd 4th5th 6th 7th 8th00 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10.43) Lateral Force Method0.30.20.11st 2nd 3rd 4th5th 6th 7th 8th00 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1B17Figure B. 16 Fragility curves <strong>of</strong> eight-storey <strong>wall</strong>-equivalent <strong>dual</strong> systems <strong>designed</strong> to DC M <strong>and</strong>PGA=0.25g for IDA, SPO <strong>and</strong> LFM analysis


Appendices0.41) Incremental Dynamic Analysis0.30.21st 2nd 3rd 4th5th 6th 7th 8th0.100 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10.42) Static Pushover Analysis0.30.20.11st 2nd 3rd 4th5th 6th 7th 8th00 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10.43) Lateral Force Method0.30.20.11st 2nd 3rd 4th5th 6th 7th 8th00 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1B18Figure B. 17 Fragility curves <strong>of</strong> eight-storey <strong>wall</strong> systems <strong>designed</strong> to DC M <strong>and</strong> PGA=0.20g for IDA,SPO <strong>and</strong> LFM analysis


Appendices0.41) Incremental Dynamic Analysis0.30.21st 2nd 3rd 4th5th 6th 7th 8th0.100 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10.42) Static Pushover Analysis0.30.20.11st 2nd 3rd 4th5th 6th 7th 8th00 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10.43) Lateral Force Method0.30.20.11st 2nd 3rd 4th5th 6th 7th 8th00 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1Figure B. 18 Fragility curves <strong>of</strong> eight-storey <strong>wall</strong> systems <strong>designed</strong> to DC M <strong>and</strong> PGA=0.25g for IDA, SPOB19<strong>and</strong> LFM analysis


AppendicesB2. Fragility curves - comparison <strong>of</strong> three methods for most critical membersAppendix B2 presents the comparison <strong>of</strong> <strong>fragility</strong> curves <strong>of</strong> the most fragile members for thethree methods <strong>of</strong> analysis. The methods <strong>of</strong> analysis include the lateral force method (LFM),nonlinear static Pushover Analysis (SPO) <strong>and</strong> the Incremental Dynamic Analysis (IDA). Thetwo damage states <strong>of</strong> yielding <strong>and</strong> ultimate are presented in each sub-figures where (a) <strong>and</strong> (b)refer to the beam yielding <strong>and</strong> ultimate state in flexure (c) <strong>and</strong> (d) refer to column yielding<strong>and</strong> ultimate state in flexure (e) <strong>and</strong> (f) refer to <strong>wall</strong> yielding <strong>and</strong> ultimate state in flexure <strong>and</strong>(g) refers to the <strong>wall</strong> ultimate state in shear.B20


Appendices0 0.2 0.4 0.6 0.8 1 1.2 1.4LFM SPO IDAFigure B. 19 Fragility curves for most critical members for results taken from LFM, SPO <strong>and</strong> IDA for afive-storey <strong>frame</strong> system <strong>designed</strong> to DC M <strong>and</strong> PGA=0.20g.B21


Appendices0 0.2 0.4 0.6 0.8 1 1.2 1.4LFM SPO IDAFigure B. 20 Fragility curves for most critical members for results taken from LFM, SPO <strong>and</strong> IDA for afive-storey <strong>frame</strong> system <strong>designed</strong> to DC M <strong>and</strong> PGA=0.25g.B22


Appendices0 0.2 0.4 0.6 0.8 1 1.2 1.4LFM SPO IDAFigure B. 21 Fragility curves for most critical members for results taken from LFM, SPO <strong>and</strong> IDA for afive-storey system <strong>designed</strong> to DC H <strong>and</strong> PGA=0.25g.B23


0.2Appendices00 0.2 0.4 0.6 0.8 1 1.2 1.4LFM (ε>1) LFM (ε=1) SPO IDAFigure B. 22 Fragility curves for most critical members for results taken from LFM with (ε>1) <strong>and</strong>without (ε=1) inelastic amplification to higher modes, SPO <strong>and</strong> IDA for a five-storey <strong>frame</strong>-equivalent<strong>dual</strong> system <strong>designed</strong> to DC M <strong>and</strong> PGA=0.20g.B24


0.2Appendices00 0.2 0.4 0.6 0.8 1 1.2 1.4LFM (ε>1) LFM (ε=1) SPO IDAFigure B. 23 Fragility curves for most critical members for results taken from LFM with (ε>1) <strong>and</strong>without (ε=1) inelastic amplification to higher modes, SPO <strong>and</strong> IDA for a five-storey <strong>frame</strong>-equivalent<strong>dual</strong> system <strong>designed</strong> to DC M <strong>and</strong> PGA=0.25g.B25


0.2Appendices00 0.2 0.4 0.6 0.8 1 1.2 1.4LFM (ε>1) LFM (ε=1) SPO IDAFigure B. 24 Fragility curves for most critical members for results taken from LFM with (ε>1) <strong>and</strong>without (ε=1) inelastic amplification to higher modes, SPO <strong>and</strong> IDA for a five-storey <strong>frame</strong>-equivalent<strong>dual</strong> system <strong>designed</strong> to DC H <strong>and</strong> PGA=0.25g.B26


0.2Appendices00 0.2 0.4 0.6 0.8 1 1.2 1.4LFM (ε>1) LFM (ε=1) SPO IDAFigure B. 25 Fragility curves for most critical members for results taken from LFM with (ε>1) <strong>and</strong>without (ε=1) inelastic amplification to higher modes, SPO <strong>and</strong> IDA for a five-storey <strong>wall</strong>-equivalent <strong>dual</strong>system <strong>designed</strong> to DC M <strong>and</strong> PGA=0.20gB27


0.2Appendices00 0.2 0.4 0.6 0.8 1 1.2 1.4LFM (ε>1) LFM (ε=1) SPO IDAFigure B. 26 Fragility curves for most critical members for results taken from LFM with (ε>1) <strong>and</strong>without (ε=1) inelastic amplification to higher modes, SPO <strong>and</strong> IDA for a five-storey <strong>wall</strong>-equivalent <strong>dual</strong>system <strong>designed</strong> to DC M <strong>and</strong> PGA=0.25gB28


0.2Appendices00 0.2 0.4 0.6 0.8 1 1.2 1.4LFM (ε>1) LFM (ε=1) SPO IDAFigure B. 27 Fragility curves for most critical members for results taken from LFM with (ε>1) <strong>and</strong>without (ε=1) inelastic amplification to higher modes, SPO <strong>and</strong> IDA for a five-storey <strong>wall</strong>-equivalent <strong>dual</strong>system <strong>designed</strong> to DC H <strong>and</strong> PGA=0.25gB29


0.2Appendices00 0.2 0.4 0.6 0.8 1 1.2 1.4LFM (ε>1) LFM (ε=1) SPO IDAFigure B. 28 Fragility curves for most critical members for results taken from LFM with (ε>1) <strong>and</strong>without (ε=1) inelastic amplification to higher modes, SPO <strong>and</strong> IDA for a five-storey <strong>wall</strong> building<strong>designed</strong> to DC M <strong>and</strong> PGA=0.20gB30


0.2Appendices00 0.2 0.4 0.6 0.8 1 1.2 1.4LFM (ε>1) LFM (ε=1) SPO IDAFigure B. 29 Fragility curves for most critical members for results taken from LFM with (ε>1) <strong>and</strong>without (ε=1) inelastic amplification to higher modes, SPO <strong>and</strong> IDA for a five-storey <strong>wall</strong> building<strong>designed</strong> to DC M <strong>and</strong> PGA=0.25gB31


0.2Appendices00 0.2 0.4 0.6 0.8 1 1.2 1.4LFM (ε>1) LFM (ε=1) SPO IDAFigure B. 30 Fragility curves for most critical members for results taken from LFM with (ε>1) <strong>and</strong>without (ε=1) inelastic amplification to higher modes, SPO <strong>and</strong> IDA for a five-storey <strong>wall</strong> building<strong>designed</strong> to DC H <strong>and</strong> PGA=0.25gB32


0.2Appendices00 0.2 0.4 0.6 0.8 1 1.2 1.4LFM (ε>1) LFM (ε=1) SPO IDAFigure B. 31 Fragility curves for most critical members for results taken from LFM with (ε>1) <strong>and</strong>without (ε=1) inelastic amplification to higher modes, SPO <strong>and</strong> IDA for a eight-storey <strong>frame</strong>-equivalent<strong>dual</strong> system <strong>designed</strong> to DC M <strong>and</strong> PGA=0.20gB33


0.2Appendices00 0.2 0.4 0.6 0.8 1 1.2 1.4LFM (ε>1) LFM (ε=1) SPO IDAFigure B. 32 Fragility curves for most critical members for results taken from LFM with (ε>1) <strong>and</strong>without (ε=1) inelastic amplification to higher modes, SPO <strong>and</strong> IDA for a eight-storey <strong>frame</strong>-equivalent<strong>dual</strong> system <strong>designed</strong> to DC M <strong>and</strong> PGA=0.25gB34


0.2Appendices00 0.2 0.4 0.6 0.8 1 1.2 1.4LFM (ε>1) LFM (ε=1) SPO IDAFigure B. 33 Fragility curves for most critical members for results taken from LFM with (ε>1) <strong>and</strong>without (ε=1) inelastic amplification to higher modes, SPO <strong>and</strong> IDA for a eight-storey <strong>wall</strong>-equivalent<strong>dual</strong> system <strong>designed</strong> to DC M <strong>and</strong> PGA=0.20gB35


0.2Appendices00 0.2 0.4 0.6 0.8 1 1.2 1.4LFM (ε>1) LFM (ε=1) SPO IDAFigure B. 34 Fragility curves for most critical members for results taken from LFM with (ε>1) <strong>and</strong>without (ε=1) inelastic amplification to higher modes, SPO <strong>and</strong> IDA for a eight-storey <strong>wall</strong>-equivalent<strong>dual</strong> system <strong>designed</strong> to DC M <strong>and</strong> PGA=0.25gB36


0.2Appendices00 0.2 0.4 0.6 0.8 1 1.2 1.4LFM (ε>1) LFM (ε=1) SPO IDAFigure B. 35 Fragility curves for most critical members for results taken from LFM with (ε>1) <strong>and</strong>without (ε=1) inelastic amplification to higher modes, SPO <strong>and</strong> IDA for a eight-storey <strong>wall</strong> building<strong>designed</strong> to DC M <strong>and</strong> PGA=0.20gB37


0.2Appendices00 0.2 0.4 0.6 0.8 1 1.2 1.4LFM (ε>1) LFM (ε=1) SPO IDAFigure B. 36 Fragility curves for most critical members for results taken from LFM with (ε>1) <strong>and</strong>without (ε=1) inelastic amplification to higher modes, SPO <strong>and</strong> IDA for a eight-storey <strong>wall</strong> building<strong>designed</strong> to DC M <strong>and</strong> PGA=0.25B38


AppendicesAPPENDIX CC1. Coefficient <strong>of</strong> variation per Intensity measureAppendix C1 presents the Coefficient <strong>of</strong> variation (CoV) <strong>of</strong> DM-dem<strong>and</strong>s as a function <strong>of</strong>intensity measure (i.e. PGA). The CoV values illustrated are taken from the nonlinear timehistoryanalysis (IDA) <strong>and</strong> are shown for each floor. On the same plot the dispersion valuesused in LFM <strong>and</strong> the SPO methods are shown in a straight line. These are the memberdispersions due to damage measure dem<strong>and</strong>s <strong>and</strong> the dispersion <strong>of</strong> the spectral value. (seeTable 7.2).In <strong>frame</strong> <strong>buildings</strong> the sub-figures (a) <strong>and</strong> (b) refer to the CoV values for beam yielding <strong>and</strong>ultimate state in flexure (c) for beam ultimate state in shear. (e) <strong>and</strong> (f) refer to the CoV valuesfor column yielding <strong>and</strong> ultimate state in flexure <strong>and</strong> (g) for column ultimate state in shear.In <strong>wall</strong>-<strong>frame</strong> <strong>buildings</strong> the sub-figures (a) <strong>and</strong> (b) refer to the CoV values for beam yielding<strong>and</strong> ultimate state in flexure <strong>and</strong> (c) <strong>and</strong> (d) for column yielding <strong>and</strong> ultimate state in flexure.(e) <strong>and</strong> (f) refer to the CoV values for <strong>wall</strong> yielding <strong>and</strong> ultimate state in flexure <strong>and</strong> (g) for<strong>wall</strong> ultimate state in shear.C1


Appendices601st 2nd 3rd 4th 5th402000 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1Figure C. 1 Coefficient <strong>of</strong> variation (CoV) <strong>of</strong> DM-dem<strong>and</strong>s determined through IDA <strong>and</strong> CoV used inSPO <strong>and</strong> LFM (straight line) for a five-storey <strong>frame</strong> system <strong>designed</strong> to DC M <strong>and</strong> PGA=0.20gC2


Appendices601st 2nd 3rd 4th 5th402000 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1Figure C. 2 Coefficient <strong>of</strong> variation (CoV) <strong>of</strong> DM-dem<strong>and</strong>s determined through IDA <strong>and</strong> CoV used inSPO <strong>and</strong> LFM (straight line) for a five-storey <strong>frame</strong> system <strong>designed</strong> to DC M <strong>and</strong> PGA=0.25gC3


Appendices601st 2nd 3rd 4th 5th402000 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1Figure C. 3 Coefficient <strong>of</strong> variation (CoV) <strong>of</strong> DM-dem<strong>and</strong>s determined through IDA <strong>and</strong> CoV used inSPO <strong>and</strong> LFM (straight line) for a five-storey <strong>frame</strong> system <strong>designed</strong> to DC H <strong>and</strong> PGA=0.25gC4


Appendices601st 2nd 3rd 4th 5th402000 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1Figure C. 4 Coefficient <strong>of</strong> variation (CoV) <strong>of</strong> DM-dem<strong>and</strong>s determined through IDA <strong>and</strong> CoV used inSPO <strong>and</strong> LFM (straight line) for a five-storey <strong>frame</strong>-equivalent <strong>dual</strong> system <strong>designed</strong> to DC M <strong>and</strong>PGA=0.20gC5


Appendices601st 2nd 3rd 4th 5th402000 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1Figure C. 5 Coefficient <strong>of</strong> variation (CoV) <strong>of</strong> DM-dem<strong>and</strong>s determined through IDA <strong>and</strong> CoV used inSPO <strong>and</strong> LFM (straight line) for a five-storey <strong>frame</strong>-equivalent <strong>dual</strong> system <strong>designed</strong> to DC M <strong>and</strong>PGA=0.25gC6


Appendices601st 2nd 3rd 4th 5th402000 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1Figure C. 6 Coefficient <strong>of</strong> variation (CoV) <strong>of</strong> DM-dem<strong>and</strong>s determined through IDA <strong>and</strong> CoV used inSPO <strong>and</strong> LFM (straight line) for a five-storey <strong>frame</strong>-equivalent <strong>dual</strong> system <strong>designed</strong> to DC H <strong>and</strong>PGA=0.25gC7


Appendices601st 2nd 3rd 4th 5th402000 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1Figure C. 7 Coefficient <strong>of</strong> variation (CoV) <strong>of</strong> DM-dem<strong>and</strong>s determined through in IDA <strong>and</strong> CoV valuesused for SPO <strong>and</strong> LFM (straight line) for a five-storey <strong>wall</strong>-equivalent <strong>dual</strong> system <strong>designed</strong> to DC M <strong>and</strong>PGA=0.20gC8


Appendices601st 2nd 3rd 4th 5th402000 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1Figure C. 8 Coefficient <strong>of</strong> variation (CoV) <strong>of</strong> DM-dem<strong>and</strong>s determined through in IDA <strong>and</strong> CoV valuesused for SPO <strong>and</strong> LFM (straight line) for a five-storey <strong>wall</strong>-equivalent <strong>dual</strong> system <strong>designed</strong> to DC M <strong>and</strong>PGA=0.25gC9


Appendices601st 2nd 3rd 4th 5th402000 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1Figure C. 9 Coefficient <strong>of</strong> variation (CoV) <strong>of</strong> DM-dem<strong>and</strong>s determined through in IDA <strong>and</strong> CoV valuesused for SPO <strong>and</strong> LFM (straight line) for a five-storey <strong>wall</strong>-equivalent <strong>dual</strong> system <strong>designed</strong> to DC H <strong>and</strong>PGA=0.25gC10


Appendices601st 2nd 3rd 4th 5th402000 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1Figure C. 10 Coefficient <strong>of</strong> variation (CoV) <strong>of</strong> DM-dem<strong>and</strong>s determined through in IDA <strong>and</strong> CoV valuesused for SPO <strong>and</strong> LFM (straight line) for a five-storey <strong>wall</strong> <strong>dual</strong> <strong>buildings</strong> <strong>designed</strong> to DC M <strong>and</strong>PGA=0.20gC11


Appendices601st 2nd 3rd 4th 5th402000 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1Figure C. 11 Coefficient <strong>of</strong> variation (CoV) <strong>of</strong> DM-dem<strong>and</strong>s determined through IDA <strong>and</strong> CoV used inSPO <strong>and</strong> LFM (straight line) for five-storey <strong>wall</strong> building <strong>designed</strong> to DC M <strong>and</strong> PGA=0.25gC12


Appendices601st 2nd 3rd 4th 5th402000 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1Figure C. 12 Coefficient <strong>of</strong> variation (CoV) <strong>of</strong> DM-dem<strong>and</strong>s determined through IDA <strong>and</strong> CoV used inSPO <strong>and</strong> LFM (straight line) for a five-storey <strong>wall</strong> <strong>dual</strong> system <strong>designed</strong> to DC H <strong>and</strong> PGA=0.25gC13


Appendices0.40.30.21st 2nd 3rd 4th5th 6th 7th 8th0.100 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1Figure C. 13 Coefficient <strong>of</strong> variation (CoV) <strong>of</strong> DM-dem<strong>and</strong>s determined through IDA <strong>and</strong> CoV used inSPO <strong>and</strong> LFM (straight line) for eight-storey <strong>frame</strong>-equivalent <strong>dual</strong> system <strong>designed</strong> to DC M <strong>and</strong>PGA=0.20gC14


Appendices0.40.30.21st 2nd 3rd 4th5th 6th 7th 8th0.100 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1Figure C. 14 Coefficient <strong>of</strong> variation (CoV) <strong>of</strong> DM-dem<strong>and</strong>s determined through IDA <strong>and</strong> CoV used inSPO <strong>and</strong> LFM (straight line) for eight-storey <strong>frame</strong>-equivalent <strong>dual</strong> system <strong>designed</strong> to DC M <strong>and</strong>PGA=0.25gC15


Appendices0.40.30.21st 2nd 3rd 4th5th 6th 7th 8th0.100 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1Figure C. 15 Coefficient <strong>of</strong> variation (CoV) <strong>of</strong> DM-dem<strong>and</strong>s determined through IDA <strong>and</strong> CoV used inSPO <strong>and</strong> LFM (straight line) for an eight-storey <strong>wall</strong>-equivalent <strong>dual</strong> system <strong>designed</strong> to DC M <strong>and</strong>PGA=0.20gC16


Appendices0.40.30.21st 2nd 3rd 4th5th 6th 7th 8th0.100 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1Figure C. 16 Coefficient <strong>of</strong> variation (CoV) <strong>of</strong> DM-dem<strong>and</strong>s determined through IDA <strong>and</strong> CoV used inSPO <strong>and</strong> LFM (straight line) for an eight-storey <strong>wall</strong>-equivalent <strong>dual</strong> system <strong>designed</strong> to DC M <strong>and</strong>PGA=0.25gC17


Appendices0.40.30.21st 2nd 3rd 4th5th 6th 7th 8th0.100 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1Figure C. 17 Coefficient <strong>of</strong> variation (CoV) <strong>of</strong> DM-dem<strong>and</strong>s determined through IDA <strong>and</strong> CoV used inC18SPO <strong>and</strong> LFM (straight line) for a eight-storey <strong>wall</strong> building <strong>designed</strong> to DC M <strong>and</strong> PGA=0.20g


Appendices0.40.30.21st 2nd 3rd 4th5th 6th 7th 8th0.100 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1Figure C. 18 Coefficient <strong>of</strong> variation (CoV) <strong>of</strong> DM-dem<strong>and</strong>s determined through IDA <strong>and</strong> CoV used inSPO <strong>and</strong> LFM (straight line) for a eight-storey <strong>wall</strong> building <strong>designed</strong> to DC M <strong>and</strong> PGA=0.25gC19


AppendicesC2. Damage indices per Intensity measureAppendix C2 presents the damage indices (ratio <strong>of</strong> the damage measure dem<strong>and</strong>s to thedamage measure capacities) taken explicitly from IDA as a function <strong>of</strong> the intensity measure(i.e. PGA). Each point on the plots represents the damage index for one record accelerogramfor either the yielding or ultimate damage state <strong>of</strong> the member. Each record is presented indifferent markers as shown in Figure C. 19. The sub-figures (a) to (e) refer to the storey <strong>of</strong> themember from the first- to fifth-storey in a five-storey building <strong>and</strong> (a) to (h) refer to the storey<strong>of</strong> the member from the first- to the eighth-storey in an eight-storey building.Figure C. 19 Legend for damage index plots where each point represents a single earthquake recordC20


AppendicesFigure C. 20 Damage indices for each floor for a beam member at yielding damage state <strong>of</strong> five-storey<strong>frame</strong> system <strong>designed</strong> to DC M <strong>and</strong> PGA=0.20gC21


AppendicesFigure C. 21 Damage indices for each floor for a beam member at ultimate damage state <strong>of</strong> five-storey<strong>frame</strong> system <strong>designed</strong> to DC M <strong>and</strong> PGA=0.20gC22


AppendicesFigure C. 22 Damage indices for each floor for a column member at yielding damage state <strong>of</strong> five-storey<strong>frame</strong> system <strong>designed</strong> to DC M <strong>and</strong> PGA=0.20gC23


AppendicesFigure C. 23 Damage indices for each floor for a column member at ultimate damage state <strong>of</strong> five-storey<strong>frame</strong> system <strong>designed</strong> to DC M <strong>and</strong> PGA=0.20gC24


AppendicesFigure C. 24 Damage indices for each floor for a beam member at shear damage state <strong>of</strong> five-storey <strong>frame</strong>system <strong>designed</strong> to DC M <strong>and</strong> PGA=0.20gC25


AppendicesFigure C. 25 Damage indices for each floor for a column member at shear damage state <strong>of</strong> five-storey<strong>frame</strong> system <strong>designed</strong> to DC M <strong>and</strong> PGA=0.20gC26


AppendicesFigure C. 26 Damage indices for each floor for a beam member at yielding damage state <strong>of</strong> five-storey<strong>frame</strong> system <strong>designed</strong> to DC M <strong>and</strong> PGA=0.25gC27


AppendicesFigure C. 27 Damage indices for each floor for a beam member at ultimate damage state <strong>of</strong> five-storey<strong>frame</strong> system <strong>designed</strong> to DC M <strong>and</strong> PGA=0.25gC28


AppendicesFigure C. 28 Damage indices for each floor for a column member at yielding damage state <strong>of</strong> five-storey<strong>frame</strong> system <strong>designed</strong> to DC M <strong>and</strong> PGA=0.25gC29


AppendicesFigure C. 29 Damage indices for each floor for a column member at ultimate damage state <strong>of</strong> five-storey<strong>frame</strong> system <strong>designed</strong> to DC M <strong>and</strong> PGA=0.25gC30


AppendicesFigure C. 30 Damage indices for each floor for a beam member at shear damage state <strong>of</strong> five-storey <strong>frame</strong>system <strong>designed</strong> to DC M <strong>and</strong> PGA=0.25gC31


AppendicesFigure C. 31 Damage indices for each floor for a column member at shear damage state <strong>of</strong> five-storey<strong>frame</strong> system <strong>designed</strong> to DC M <strong>and</strong> PGA=0.25gC32


AppendicesFigure C. 32 Damage indices for each floor for a beam member at yielding damage state <strong>of</strong> five-storey<strong>frame</strong> system <strong>designed</strong> to DC H <strong>and</strong> PGA=0.25gC33


AppendicesFigure C. 33 Damage indices for each floor for a beam member at ultimate damage state <strong>of</strong> five-storey<strong>frame</strong> system <strong>designed</strong> to DC H <strong>and</strong> PGA=0.25gC34


AppendicesFigure C. 34 Damage indices for each floor for a column member at yielding damage state <strong>of</strong> five-storey<strong>frame</strong> system <strong>designed</strong> to DC H <strong>and</strong> PGA=0.25gC35


AppendicesFigure C. 35 Damage indices for each floor for a column member at ultimate damage state <strong>of</strong> five-storey<strong>frame</strong> system <strong>designed</strong> to DC H <strong>and</strong> PGA=0.25gC36


AppendicesFigure C. 36 Damage indices for each floor for a beam member at shear damage state <strong>of</strong> five-storey <strong>frame</strong>system <strong>designed</strong> to DC H <strong>and</strong> PGA=0.25gC37


AppendicesFigure C. 37 Damage indices for each floor for a column member at shear damage state <strong>of</strong> five-storey<strong>frame</strong> system <strong>designed</strong> to DC H <strong>and</strong> PGA=0.25gC38


AppendicesFigure C. 38 Damage indices for each floor for a beam member at yielding damage state <strong>of</strong> five-storey<strong>frame</strong>-equivalent <strong>dual</strong> system <strong>designed</strong> to DC M <strong>and</strong> PGA=0.20gC39


AppendicesFigure C. 39 Damage indices for each floor for a beam member at ultimate damage state <strong>of</strong> five-storey<strong>frame</strong>-equivalent <strong>dual</strong> system <strong>designed</strong> to DC M <strong>and</strong> PGA=0.20gC40


AppendicesFigure C. 40 Damage indices for each floor for a column member at yielding damage state <strong>of</strong> five-storey<strong>frame</strong>-equivalent <strong>dual</strong> system <strong>designed</strong> to DC M <strong>and</strong> PGA=0.20gC41


AppendicesFigure C. 41 Damage indices for each floor for a column member at ultimate damage state <strong>of</strong> five-storey<strong>frame</strong>-equivalent <strong>dual</strong> system <strong>designed</strong> to DC M <strong>and</strong> PGA=0.20gC42


AppendicesFigure C. 42 Damage indices for each floor for a <strong>wall</strong> member at a) yielding b) ultimate c) shear ultimatedamage state <strong>of</strong> five-storey <strong>frame</strong>-equivalent system <strong>designed</strong> to DC M <strong>and</strong> PGA=0.20gC43


AppendicesFigure C. 43 Damage indices for each floor for a beam member at yielding damage state <strong>of</strong> five-storey<strong>frame</strong>-equivalent <strong>dual</strong> system <strong>designed</strong> to DC M <strong>and</strong> PGA=0.25gC44


AppendicesFigure C. 44 Damage indices for each floor for a beam member at ultimate damage state <strong>of</strong> five-storey<strong>frame</strong>-equivalent <strong>dual</strong> system <strong>designed</strong> to DC M <strong>and</strong> PGA=0.25gC45


AppendicesFigure C. 45 Damage indices for each floor for a column member at yielding damage state <strong>of</strong> five-storey<strong>frame</strong>-equivalent <strong>dual</strong> system <strong>designed</strong> to DC M <strong>and</strong> PGA=0.25gC46


AppendicesFigure C. 46 Damage indices for each floor for a column member at ultimate damage state <strong>of</strong> five-storey<strong>frame</strong>-equivalent <strong>dual</strong> system <strong>designed</strong> to DC M <strong>and</strong> PGA=0.25gC47


AppendicesFigure C. 47 Damage indices for each floor for a <strong>wall</strong> member at a) yielding b) ultimate c) shear ultimatedamage state <strong>of</strong> five-storey <strong>frame</strong>-equivalent system <strong>designed</strong> to DC M <strong>and</strong> PGA=0.25gC48


AppendicesFigure C. 48 Damage indices for each floor for a beam member at yielding damage state <strong>of</strong> five-storey<strong>frame</strong>-equivalent <strong>dual</strong> system <strong>designed</strong> to DC H <strong>and</strong> PGA=0.25gC49


AppendicesFigure C. 49 Damage indices for each floor for a beam member at ultimate damage state <strong>of</strong> five-storey<strong>frame</strong>-equivalent <strong>dual</strong> system <strong>designed</strong> to DC H <strong>and</strong> PGA=0.25gC50


AppendicesFigure C. 50 Damage indices for each floor for a column member at yielding damage state <strong>of</strong> five-storey<strong>frame</strong>-equivalent <strong>dual</strong> system <strong>designed</strong> to DC H <strong>and</strong> PGA=0.25gC51


AppendicesFigure C. 51 Damage indices for each floor for a column member at ultimate damage state <strong>of</strong> five-storey<strong>frame</strong>-equivalent <strong>dual</strong> system <strong>designed</strong> to DC H <strong>and</strong> PGA=0.25gC52


AppendicesFigure C. 52 Damage indices for each floor for a <strong>wall</strong> member at a) yielding b) ultimate c) shear ultimatedamage state <strong>of</strong> five-storey <strong>frame</strong>-equivalent system <strong>designed</strong> to DC H <strong>and</strong> PGA=0.25gC53


AppendicesFigure C. 53 Damage indices for each floor for a beam member at yielding damage state <strong>of</strong> five-storey<strong>wall</strong>-equivalent <strong>dual</strong> system <strong>designed</strong> to DC M <strong>and</strong> PGA=0.20gC54


AppendicesFigure C. 54 Damage indices for each floor for a beam member at ultimate damage state <strong>of</strong> five-storey<strong>wall</strong>-equivalent <strong>dual</strong> system <strong>designed</strong> to DC M <strong>and</strong> PGA=0.20gC55


AppendicesFigure C. 55 Damage indices for each floor for a column member at yielding damage state <strong>of</strong> five-storey<strong>wall</strong>-equivalent <strong>dual</strong> system <strong>designed</strong> to DC M <strong>and</strong> PGA=0.20gC56


AppendicesFigure C. 56 Damage indices for each floor for a column member at ultimate damage state <strong>of</strong> five-storey<strong>wall</strong>-equivalent <strong>dual</strong> system <strong>designed</strong> to DC M <strong>and</strong> PGA=0.20gC57


AppendicesFigure C. 57 Damage indices for each floor for a <strong>wall</strong> member at a) yielding b) ultimate c) shear ultimatedamage state <strong>of</strong> five-storey <strong>wall</strong>-equivalent system <strong>designed</strong> to DC M <strong>and</strong> PGA=0.20gC58


AppendicesFigure C. 58 Damage indices for each floor for a beam member at yielding damage state <strong>of</strong> five-storey<strong>wall</strong>-equivalent <strong>dual</strong> system <strong>designed</strong> to DC M <strong>and</strong> PGA=0.25gC59


AppendicesFigure C. 59 Damage indices for each floor for a beam member at ultimate damage state <strong>of</strong> five-storey<strong>wall</strong>-equivalent <strong>dual</strong> system <strong>designed</strong> to DC M <strong>and</strong> PGA=0.25gC60


AppendicesFigure C. 60 Damage indices for each floor for a column member at yielding damage state <strong>of</strong> five-storey<strong>wall</strong>-equivalent <strong>dual</strong> system <strong>designed</strong> to DC M <strong>and</strong> PGA=0.25gC61


AppendicesFigure C. 61 Damage indices for each floor for a column member at ultimate damage state <strong>of</strong> five-storey<strong>wall</strong>-equivalent <strong>dual</strong> system <strong>designed</strong> to DC M <strong>and</strong> PGA=0.25gC62


AppendicesFigure C. 62 Damage indices for each floor for a <strong>wall</strong> member at a) yielding b) ultimate c) shear ultimatedamage state <strong>of</strong> five-storey <strong>wall</strong>-equivalent system <strong>designed</strong> to DC M <strong>and</strong> PGA=0.25gC63


AppendicesFigure C. 63 Damage indices for each floor for a beam member at yielding damage state <strong>of</strong> five-storey<strong>wall</strong>-equivalent <strong>dual</strong> system <strong>designed</strong> to DC H <strong>and</strong> PGA=0.25gC64


AppendicesFigure C. 64 Damage indices for each floor for a beam member at ultimate damage state <strong>of</strong> five-storey<strong>wall</strong> -equivalent <strong>dual</strong> system <strong>designed</strong> to DC H <strong>and</strong> PGA=0.25gC65


AppendicesFigure C. 65 Damage indices for each floor for a column member at yielding damage state <strong>of</strong> five-storey<strong>wall</strong> -equivalent <strong>dual</strong> system <strong>designed</strong> to DC H <strong>and</strong> PGA=0.25gC66


AppendicesFigure C. 66 Damage indices for each floor for a column member at ultimate damage state <strong>of</strong> five-storey<strong>wall</strong> -equivalent <strong>dual</strong> system <strong>designed</strong> to DC H <strong>and</strong> PGA=0.25gC67


AppendicesFigure C. 67 Damage indices for each floor for a <strong>wall</strong> member at a) yielding b) ultimate c) shear ultimatedamage state <strong>of</strong> five-storey <strong>wall</strong> -equivalent system <strong>designed</strong> to DC H <strong>and</strong> PGA=0.25gC68


AppendicesFigure C. 68 Damage indices for each floor for a beam member at yielding damage state <strong>of</strong> five-storey <strong>wall</strong>system <strong>designed</strong> to DC M <strong>and</strong> PGA=0.20gC69


AppendicesFigure C. 69 Damage indices for each floor for a beam member at ultimate damage state <strong>of</strong> five-storey<strong>wall</strong> system <strong>designed</strong> to DC M <strong>and</strong> PGA=0.20gC70


AppendicesFigure C. 70 Damage indices for each floor for a column member at yielding damage state <strong>of</strong> five-storey<strong>wall</strong> system <strong>designed</strong> to DC M <strong>and</strong> PGA=0.20gC71


AppendicesFigure C. 71 Damage indices for each floor for a column member at ultimate damage state <strong>of</strong> five-storey<strong>wall</strong> system <strong>designed</strong> to DC M <strong>and</strong> PGA=0.20gC72


AppendicesFigure C. 72 Damage indices for each floor for a <strong>wall</strong> member at a) yielding b) ultimate c) shear ultimatedamage state <strong>of</strong> five-storey <strong>wall</strong> system <strong>designed</strong> to DC M <strong>and</strong> PGA=0.20gC73


AppendicesFigure C. 73 Damage indices for each floor for a beam member at yielding damage state <strong>of</strong> five-storey <strong>wall</strong>system <strong>designed</strong> to DC M <strong>and</strong> PGA=0.25gC74


AppendicesFigure C. 74 Damage indices for each floor for a beam member at ultimate damage state <strong>of</strong> five-storey<strong>wall</strong> system <strong>designed</strong> to DC M <strong>and</strong> PGA=0.25gC75


AppendicesFigure C. 75 Damage indices for each floor for a column member at yielding damage state <strong>of</strong> five-storey<strong>wall</strong> system <strong>designed</strong> to DC M <strong>and</strong> PGA=0.25gC76


AppendicesFigure C. 76 Damage indices for each floor for a column member at ultimate damage state <strong>of</strong> five-storey<strong>wall</strong> system <strong>designed</strong> to DC M <strong>and</strong> PGA=0.25gC77


AppendicesFigure C. 77 Damage indices for each floor for a <strong>wall</strong> member at a) yielding b) ultimate c) shear ultimatedamage state <strong>of</strong> five-storey <strong>wall</strong> system <strong>designed</strong> to DC M <strong>and</strong> PGA=0.25gC78


AppendicesFigure C. 78 Damage indices for each floor for a beam member at yielding damage state <strong>of</strong> five-storey <strong>wall</strong>system <strong>designed</strong> to DC H <strong>and</strong> PGA=0.25gC79


AppendicesFigure C. 79 Damage indices for each floor for a beam member at ultimate damage state <strong>of</strong> five-storey<strong>wall</strong> system <strong>designed</strong> to DC H <strong>and</strong> PGA=0.25gC80


AppendicesFigure C. 80 Damage indices for each floor for a column member at yielding damage state <strong>of</strong> five-storey<strong>wall</strong> system <strong>designed</strong> to DC H <strong>and</strong> PGA=0.25gC81


AppendicesFigure C. 81 Damage indices for each floor for a column member at ultimate damage state <strong>of</strong> five-storey<strong>wall</strong> system <strong>designed</strong> to DC H <strong>and</strong> PGA=0.25gC82


AppendicesFigure C. 82 Damage indices for each floor for a <strong>wall</strong> member at a) yielding b) ultimate c) shear ultimatedamage state <strong>of</strong> five-storey <strong>wall</strong> system <strong>designed</strong> to DC H <strong>and</strong> PGA=0.25gC83


AppendicesFigure C. 83 Damage indices for each floor for a beam member at yielding damage state <strong>of</strong> eight-storeyC84<strong>frame</strong>-equivalent system <strong>designed</strong> to DC M <strong>and</strong> PGA=0.20g


AppendicesFigure C. 84 Damage indices for each floor for a beam member at ultimate damage state <strong>of</strong> eight-storeyC85<strong>frame</strong>-equivalent system <strong>designed</strong> to DC M <strong>and</strong> PGA=0.20g


AppendicesFigure C. 85 Damage indices for each floor for a column member at yielding damage state <strong>of</strong> eight-storeyC86<strong>frame</strong>-equivalent system <strong>designed</strong> to DC M <strong>and</strong> PGA=0.20g


AppendicesFigure C. 86 Damage indices for each floor for a column member at ultimate damage state <strong>of</strong> eight-storeyC87<strong>frame</strong>-equivalent system <strong>designed</strong> to DC M <strong>and</strong> PGA=0.20g


AppendicesFigure C. 87 Damage indices for each floor for a <strong>wall</strong> member at a) yielding b) ultimate c) shear ultimatedamage state <strong>of</strong> five-storey <strong>frame</strong>-equivalent system <strong>designed</strong> to DC M <strong>and</strong> PGA=0.20gC88


AppendicesFigure C. 88 Damage indices for each floor for a beam member at yielding damage state <strong>of</strong> eight-storeyC89<strong>frame</strong>-equivalent system <strong>designed</strong> to DC M <strong>and</strong> PGA=0.25g


AppendicesFigure C. 89 Damage indices for each floor for a beam member at ultimate damage state <strong>of</strong> eight-storeyC90<strong>frame</strong>-equivalent system <strong>designed</strong> to DC M <strong>and</strong> PGA=0.25g


AppendicesFigure C. 90 Damage indices for each floor for a column member at yielding damage state <strong>of</strong> eight-storeyC91<strong>frame</strong>-equivalent system <strong>designed</strong> to DC M <strong>and</strong> PGA=0.25g


AppendicesFigure C. 91 Damage indices for each floor for a column member at ultimate damage state <strong>of</strong> eight-storeyC92<strong>frame</strong>-equivalent system <strong>designed</strong> to DC M <strong>and</strong> PGA=0.25g


AppendicesFigure C. 92 Damage indices for each floor for a <strong>wall</strong> member at a) yielding b) ultimate c) shear ultimatedamage state <strong>of</strong> eight-storey <strong>frame</strong>-equivalent system <strong>designed</strong> to DC M <strong>and</strong> PGA=0.25gC93


AppendicesFigure C. 93 Damage indices for each floor for a beam member at yielding damage state <strong>of</strong> eight-storeyC94<strong>wall</strong>-equivalent system <strong>designed</strong> to DC M <strong>and</strong> PGA=0.20g


AppendicesFigure C. 94 Damage indices for each floor for a beam member at ultimate damage state <strong>of</strong> eight-storeyC95<strong>wall</strong>-equivalent system <strong>designed</strong> to DC M <strong>and</strong> PGA=0.20g


AppendicesFigure C. 95 Damage indices for each floor for a column member at yielding damage state <strong>of</strong> eight-storeyC96<strong>wall</strong>-equivalent system <strong>designed</strong> to DC M <strong>and</strong> PGA=0.20g


AppendicesFigure C. 96 Damage indices for each floor for a column member at ultimate damage state <strong>of</strong> eight-storeyC97<strong>wall</strong>-equivalent system <strong>designed</strong> to DC M <strong>and</strong> PGA=0.20g


AppendicesFigure C. 97 Damage indices for each floor for a <strong>wall</strong> member at a) yielding b) ultimate c) shear ultimatedamage state <strong>of</strong> eight-storey <strong>wall</strong>-equivalent system <strong>designed</strong> to DC M <strong>and</strong> PGA=0.20gC98


AppendicesFigure C. 98 Damage indices for each floor for a beam member at yielding damage state <strong>of</strong> eight-storeyC99<strong>wall</strong>-equivalent system <strong>designed</strong> to DC M <strong>and</strong> PGA=0.25g


AppendicesFigure C. 99 Damage indices for each floor for a beam member at ultimate damage state <strong>of</strong> eight-storeyC100<strong>wall</strong>-equivalent system <strong>designed</strong> to DC M <strong>and</strong> PGA=0.25g


AppendicesFigure C. 100 Damage indices for each floor for a column member at yielding damage state <strong>of</strong> eight-storeyC101<strong>wall</strong>-equivalent system <strong>designed</strong> to DC M <strong>and</strong> PGA=0.25g


AppendicesC102Figure C. 101 Damage indices for each floor for a column member at ultimate damage state <strong>of</strong> eightstorey<strong>wall</strong>-equivalent system <strong>designed</strong> to DC M <strong>and</strong> PGA=0.25g


AppendicesFigure C. 102 Damage indices for each floor for a <strong>wall</strong> member at a) yielding b) ultimate c) shear ultimatedamage state <strong>of</strong> eight-storey <strong>wall</strong>-equivalent system <strong>designed</strong> to DC M <strong>and</strong> PGA=0.25gC103


AppendicesFigure C. 103 Damage indices for each floor for a beam member at yielding damage state <strong>of</strong> eight-storeyC104<strong>wall</strong> system <strong>designed</strong> to DC M <strong>and</strong> PGA=0.20g


AppendicesFigure C. 104 Damage indices for each floor for a beam member at ultimate damage state <strong>of</strong> eight-storeyC105<strong>wall</strong> system <strong>designed</strong> to DC M <strong>and</strong> PGA=0.20g


AppendicesFigure C. 105 Damage indices for each floor for a column member at yielding damage state <strong>of</strong> eight-storeyC106<strong>wall</strong> system <strong>designed</strong> to DC M <strong>and</strong> PGA=0.20g


AppendicesC107Figure C. 106 Damage indices for each floor for a column member at ultimate damage state <strong>of</strong> eightstorey<strong>wall</strong> system <strong>designed</strong> to DC M <strong>and</strong> PGA=0.20g


AppendicesFigure C. 107 Damage indices for each floor for a <strong>wall</strong> member at a) yielding b) ultimate c) shear ultimatedamage state <strong>of</strong> eight-storey <strong>wall</strong>-equivalent system <strong>designed</strong> to DC M <strong>and</strong> PGA=0.20gC108


AppendicesFigure C. 108 Damage indices for each floor for a beam member at yielding damage state <strong>of</strong> eight-storeyC109<strong>wall</strong> system <strong>designed</strong> to DC M <strong>and</strong> PGA=0.25g


AppendicesFigure C. 109 Damage indices for each floor for a beam member at ultimate damage state <strong>of</strong> eight-storeyC110<strong>wall</strong> – system <strong>designed</strong> to DC M <strong>and</strong> PGA=0.25g


AppendicesFigure C. 110 Damage indices for each floor for a column member at yielding damage state <strong>of</strong> eight-storeyC111<strong>wall</strong> system <strong>designed</strong> to DC M <strong>and</strong> PGA=0.25g


AppendicesC112Figure C. 111 Damage indices for each floor for a column member at ultimate damage state <strong>of</strong> eightstorey<strong>wall</strong> system <strong>designed</strong> to DC M <strong>and</strong> PGA=0.25g


AppendicesFigure C. 112 Damage indices for each floor for a <strong>wall</strong> member at a) yielding b) ultimate c) shear ultimatedamage state <strong>of</strong> eight-storey <strong>wall</strong> system <strong>designed</strong> to DC M <strong>and</strong> PGA=0.25gC113


MSc Dissertation 2013 <strong>Seismic</strong> <strong>fragility</strong> <strong>of</strong> <strong>RC</strong> <strong>frame</strong> <strong>and</strong> <strong>wall</strong>-<strong>frame</strong> <strong>dual</strong> <strong>buildings</strong> <strong>designed</strong> to EN-Eurocodes Kyriakos Antoniou

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